439 lines
19 KiB
Plaintext
439 lines
19 KiB
Plaintext
ChatGPT Antwort
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##################################################################################################################
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Before I begin coding, I’d like to clarify a few points to ensure that my solution meets your requirements as closely as possible:
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1. **Operations to Implement:**
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- Should the red-black tree support only insertion (and the associated balancing/fix-ups) or should it also include deletion and search operations?
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- Are there any additional operations you expect the tree to have?
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2. **Visualization Details:**
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- Do you want a static visualization of the final tree (using Matplotlib to display nodes and edges) or an animated step-by-step visualization that shows the tree as nodes are inserted and the tree is rebalanced?
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- Do you have any preferences for the layout of the tree (e.g., spacing, orientation) or styling details for the nodes and edges (e.g., displaying node values, coloring nodes as red or black)?
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3. **Input and Testing:**
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- Should the program include a predefined set of insertions (e.g., a list of numbers) to demonstrate the tree, or would you prefer to have a way for the user to input values interactively?
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4. **Assumptions and Explanations:**
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- Would you like detailed inline comments and a separate explanation of the reasoning behind the implementation, or is a brief summary sufficient?
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Once I have your clarifications on these points, I can proceed by breaking the problem into sub-problems and then combining everything into one complete solution. Please let me know your preferences!
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####################################################################################################################
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Below is a summary of my approach to solving the problem:
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1. **Dividing the Problem into Sub-Problems:**
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- **Data Structures and Core Classes:**
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- **Node Class:**
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Each node will store a value, its color (red or black), and pointers to its left child, right child, and parent.
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- **Tree Class:**
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This class will maintain the root of the red-black tree and will include methods for insertion, deletion, and search. It will also implement the required rotations (left and right) and fix-up procedures to maintain the red-black tree properties after insertions or deletions.
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- **Red-Black Tree Operations:**
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- **Insertion:**
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We will perform a standard BST insertion and then run an insertion fix-up to restore the red-black properties (e.g., ensuring that no red node has a red child and that all paths have the same black height).
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- **Deletion:**
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Deletion in a red-black tree is more involved. We will remove the node as in a standard BST deletion and then execute a deletion fix-up procedure to maintain the tree’s properties.
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- **Search:**
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A standard BST search will be used to locate nodes.
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- **Dynamic Visualization Using Matplotlib:**
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- We will create a function that computes the positions of nodes so that the tree grows vertically (with the root at the top and children below).
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- Each node will be drawn as a circle with its value inside, colored red or black based on the node’s color.
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- The visualization will be dynamic, updating after each operation (insert, delete, or search) using Matplotlib’s animation or interactive update capabilities.
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- We plan to animate the steps (for example, after each insertion or deletion) to provide a clear view of how the tree rebalances itself.
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2. **Assumptions and Considerations:**
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- **Operations:**
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The tree will support insert, delete, and search operations.
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- **Visualization:**
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The tree will be displayed with a vertical layout. Nodes will be clearly colored (red or black) and annotated with their values. We will implement dynamic updating of the tree visualization after each operation.
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- **Testing:**
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A set of arbitrary numbers will be inserted into the tree for demonstration purposes.
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- **Limitations:**
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- The deletion algorithm for red-black trees can be quite complex. While we’ll implement the standard approach, there might be edge cases that need more extensive testing in a production scenario.
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- The dynamic visualization will depend on Matplotlib’s capabilities. In some environments, interactive animations may behave differently or require specific backends.
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- **Environment:**
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We assume that Python and Matplotlib (and any necessary libraries) are available and up-to-date.
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3. **Combining the Components:**
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- The final solution will consist of the Node and Tree classes with all the required operations.
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- After performing operations (insertion, deletion, search), the tree’s state will be re-plotted dynamically to show the current structure.
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- The complete solution will be integrated into a single Python script that can be run to see the red-black tree in action.
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If everything looks good with this plan, I’ll proceed to implement the code accordingly.
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##################################################################################################################
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Below is the complete Python script that implements a red‐black tree with insert, delete, and search operations and uses Matplotlib to dynamically visualize the tree after each operation. (The code uses a sentinel NIL node to simplify the red–black tree algorithms.) You can run the script to see a vertical tree that updates after each insertion or deletion.
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To explain briefly (outside of inline comments):
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1. We define a **Node** class that holds a value, its color ("red" or "black"), and pointers to its left child, right child, and parent.
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2. The **RedBlackTree** class creates a NIL sentinel (a black node with no value) and sets the tree’s root to this NIL. It provides methods for insertion (with an insert_fixup), deletion (with transplant, minimum, and delete_fixup helper methods), and search. The standard red–black tree algorithms from textbooks (e.g. CLRS) are used.
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3. For dynamic visualization, the tree has a method called **draw_tree**. It recursively computes positions for nodes (assigning x coordinates by in‐order sequence and y coordinates by depth, with the root at the top) and then uses Matplotlib to draw lines for edges and circles (colored red or black) for nodes, with the node’s value written in the circle. After each tree operation, draw_tree is called and plt.pause is used to animate the changes.
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4. In the main section a set of arbitrary numbers is inserted into the tree (with updates shown), then a couple of deletions are performed, and a search is done.
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You can modify the numbers and operations as desired.
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Below is the complete code:
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---
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```python
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import matplotlib.pyplot as plt
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import time
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# -------------------------------
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# Node Class
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# -------------------------------
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class Node:
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def __init__(self, value, color="red", left=None, right=None, parent=None):
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self.value = value # Stored value (None for the NIL node)
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self.color = color # "red" or "black"
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self.left = left # Left child
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self.right = right # Right child
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self.parent = parent # Parent pointer
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# -------------------------------
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# Red-Black Tree Class
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# -------------------------------
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class RedBlackTree:
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def __init__(self):
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# Create the NIL sentinel node. This is used in place of None.
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self.NIL = Node(value=None, color="black")
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self.NIL.left = self.NIL.right = self.NIL.parent = self.NIL
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self.root = self.NIL
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# ---------------------------
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# Left Rotate x around its right child y.
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# ---------------------------
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def left_rotate(self, x):
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y = x.right
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x.right = y.left
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if y.left != self.NIL:
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y.left.parent = x
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y.parent = x.parent
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if x.parent == self.NIL:
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self.root = y
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elif x == x.parent.left:
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x.parent.left = y
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else:
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x.parent.right = y
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y.left = x
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x.parent = y
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# ---------------------------
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# Right Rotate x around its left child y.
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# ---------------------------
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def right_rotate(self, x):
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y = x.left
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x.left = y.right
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if y.right != self.NIL:
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y.right.parent = x
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y.parent = x.parent
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if x.parent == self.NIL:
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self.root = y
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elif x == x.parent.right:
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x.parent.right = y
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else:
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x.parent.left = y
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y.right = x
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x.parent = y
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# ---------------------------
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# Insert a new value into the red-black tree.
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# ---------------------------
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def insert(self, value):
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# Create new node with left and right children set to NIL.
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z = Node(value=value, color="red", left=self.NIL, right=self.NIL, parent=self.NIL)
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y = self.NIL
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x = self.root
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# Standard BST insertion.
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while x != self.NIL:
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y = x
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if z.value < x.value:
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x = x.left
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else:
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x = x.right
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z.parent = y
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if y == self.NIL:
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self.root = z
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elif z.value < y.value:
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y.left = z
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else:
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y.right = z
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# Fix up any red-black violations.
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self.insert_fixup(z)
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# ---------------------------
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# Restore red-black properties after insertion.
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# ---------------------------
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def insert_fixup(self, z):
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while z.parent.color == "red":
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if z.parent == z.parent.parent.left:
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y = z.parent.parent.right # uncle
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if y.color == "red":
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# Case 1: Uncle is red.
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z.parent.color = "black"
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y.color = "black"
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z.parent.parent.color = "red"
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z = z.parent.parent
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else:
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if z == z.parent.right:
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# Case 2: z is right child.
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z = z.parent
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self.left_rotate(z)
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# Case 3: z is left child.
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z.parent.color = "black"
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z.parent.parent.color = "red"
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self.right_rotate(z.parent.parent)
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else:
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# Mirror image of the above code.
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y = z.parent.parent.left
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if y.color == "red":
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z.parent.color = "black"
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y.color = "black"
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z.parent.parent.color = "red"
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z = z.parent.parent
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else:
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if z == z.parent.left:
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z = z.parent
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self.right_rotate(z)
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z.parent.color = "black"
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z.parent.parent.color = "red"
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self.left_rotate(z.parent.parent)
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self.root.color = "black"
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# ---------------------------
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# Transplant: Replace subtree rooted at u with subtree rooted at v.
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# ---------------------------
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def transplant(self, u, v):
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if u.parent == self.NIL:
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self.root = v
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elif u == u.parent.left:
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u.parent.left = v
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else:
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u.parent.right = v
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v.parent = u.parent
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# ---------------------------
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# Find the minimum node starting from a given node.
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# ---------------------------
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def minimum(self, node):
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while node.left != self.NIL:
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node = node.left
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return node
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# ---------------------------
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# Delete a node with a given value from the tree.
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# ---------------------------
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def delete_value(self, value):
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node = self.search(value)
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if node == self.NIL:
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print(f"Value {value} not found in the tree.")
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return
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self.delete_node(node)
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# ---------------------------
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# Delete a node z from the tree.
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# ---------------------------
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def delete_node(self, z):
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y = z
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y_original_color = y.color
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if z.left == self.NIL:
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x = z.right
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self.transplant(z, z.right)
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elif z.right == self.NIL:
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x = z.left
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self.transplant(z, z.left)
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else:
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y = self.minimum(z.right)
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y_original_color = y.color
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x = y.right
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if y.parent == z:
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x.parent = y
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else:
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self.transplant(y, y.right)
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y.right = z.right
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y.right.parent = y
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self.transplant(z, y)
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y.left = z.left
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y.left.parent = y
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y.color = z.color
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if y_original_color == "black":
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self.delete_fixup(x)
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# ---------------------------
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# Restore red-black properties after deletion.
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# ---------------------------
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def delete_fixup(self, x):
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while x != self.root and x.color == "black":
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if x == x.parent.left:
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w = x.parent.right
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if w.color == "red":
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w.color = "black"
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x.parent.color = "red"
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self.left_rotate(x.parent)
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w = x.parent.right
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if w.left.color == "black" and w.right.color == "black":
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w.color = "red"
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x = x.parent
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else:
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if w.right.color == "black":
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w.left.color = "black"
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w.color = "red"
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self.right_rotate(w)
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w = x.parent.right
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w.color = x.parent.color
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x.parent.color = "black"
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w.right.color = "black"
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self.left_rotate(x.parent)
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x = self.root
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else:
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# Mirror image of above code.
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w = x.parent.left
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if w.color == "red":
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w.color = "black"
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x.parent.color = "red"
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self.right_rotate(x.parent)
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w = x.parent.left
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if w.right.color == "black" and w.left.color == "black":
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w.color = "red"
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x = x.parent
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else:
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if w.left.color == "black":
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w.right.color = "black"
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w.color = "red"
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self.left_rotate(w)
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w = x.parent.left
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w.color = x.parent.color
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x.parent.color = "black"
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w.left.color = "black"
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self.right_rotate(x.parent)
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x = self.root
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x.color = "black"
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# ---------------------------
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# Search for a node with a given value.
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# ---------------------------
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def search(self, value):
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current = self.root
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while current != self.NIL and current.value != value:
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if value < current.value:
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current = current.left
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else:
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current = current.right
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return current
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# ---------------------------
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# Dynamic visualization of the tree using Matplotlib.
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# The tree is drawn vertically with the root at the top.
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# ---------------------------
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def draw_tree(self):
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plt.clf() # Clear current figure
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positions = {} # Dictionary mapping node -> (x, y)
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# Helper function to perform an in-order traversal and assign positions.
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def compute_positions(node, depth, pos_dict, x_counter):
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if node == self.NIL:
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return
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compute_positions(node.left, depth + 1, pos_dict, x_counter)
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# x_counter[0] holds the current horizontal position.
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pos_dict[node] = (x_counter[0], -depth)
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x_counter[0] += 1
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compute_positions(node.right, depth + 1, pos_dict, x_counter)
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compute_positions(self.root, 0, positions, [0])
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ax = plt.gca()
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# Draw the edges (lines between parent and child).
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for node, (x, y) in positions.items():
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if node.left != self.NIL:
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child_pos = positions[node.left]
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ax.plot([x, child_pos[0]], [y, child_pos[1]], 'k-')
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if node.right != self.NIL:
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child_pos = positions[node.right]
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ax.plot([x, child_pos[0]], [y, child_pos[1]], 'k-')
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# Draw the nodes (circles with text).
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for node, (x, y) in positions.items():
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circle = plt.Circle((x, y), 0.3, color=node.color, ec="black", zorder=2)
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ax.add_patch(circle)
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# For readability: if the node is black, write white text; otherwise black text.
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text_color = "white" if node.color == "black" else "black"
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ax.text(x, y, str(node.value), ha="center", va="center", color=text_color, zorder=3)
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ax.set_aspect(1.0)
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ax.axis("off")
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plt.pause(0.5) # Pause to animate the update
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# -------------------------------
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# Main testing and dynamic visualization
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# -------------------------------
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if __name__ == "__main__":
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tree = RedBlackTree()
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# Turn on interactive mode for dynamic plotting.
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plt.ion()
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# Insert some arbitrary numbers into the tree.
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numbers_to_insert = [10, 20, 30, 15, 25, 5, 1]
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print("Inserting values:", numbers_to_insert)
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for num in numbers_to_insert:
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tree.insert(num)
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tree.draw_tree() # Update the visualization after each insertion
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time.sleep(0.5) # Optional: add extra pause for clarity
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# Delete a couple of nodes.
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print("Deleting value 20")
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tree.delete_value(20)
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tree.draw_tree()
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time.sleep(0.5)
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print("Deleting value 10")
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tree.delete_value(10)
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tree.draw_tree()
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time.sleep(0.5)
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# Search for a value.
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search_value = 15
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result = tree.search(search_value)
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if result != tree.NIL:
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print(f"Search: Found node with value {result.value}")
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else:
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print(f"Search: Value {search_value} not found.")
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# Keep the final plot open.
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plt.ioff()
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plt.show()
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```
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---
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### How to Run
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1. Make sure you have Python 3 and Matplotlib installed (e.g., via `pip install matplotlib`).
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2. Save the code above into a file (for example, `red_black_tree.py`).
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3. Run the script with:
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`python red_black_tree.py`
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4. A Matplotlib window will open and update dynamically as nodes are inserted and deleted. The console will also print the operations and search result.
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### Limitations and Considerations
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- **Deletion Complexity:** Red–black tree deletion is intricate. While this implementation follows the standard algorithm, extensive testing in production systems is recommended.
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- **Visualization Layout:** The in-order traversal assigns x coordinates based solely on node order. For very unbalanced trees, the layout might be uneven.
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- **Dynamic Updates:** The dynamic visualization uses Matplotlib’s interactive mode (`plt.ion()`) and `plt.pause()`, which may behave differently in some environments or IDEs.
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Feel free to adjust timing, styling, or add further error checking as needed.
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