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# Red Black Trees
📗 Red Black Trees (RBT) are Binary Search Trees (BST) that stay balanced (self balancing).
➩ Each node is either red or black. (null children are considered black.)
➩ The root node is black.
➩ No red nodes have red children.
➩ Every path from root to a null child has the same number of black nodes (black height of the tree).
📗 Visualization: Link.
# RBT Example
ID:📗 [1 point] Recolor the BST so that it is a RBT.
# RBT Balance
📗 Compare the longest and shortest paths:
➩ Identical black lengths \(H_{b}\).
➩ Shortest paths have \(H_{b} = O\left(\log\left(N\right)\right)\) lengths.
➩ Longest paths have \(2 H_{b} = O\left(2 \log\left(N\right)\right) = O\left(\log\left(N\right)\right)\) lengths.
➩ \(H = O\left(\log\left(N\right)\right)\) for RBTs.
# Inserting into a RBT
📗 Insert new value using BST insertion algorithm.
📗 Color the new node red.
📗 Check RBT properties and restore if necessary.
➩ The root node is black.
➩ No red nodes have red children.
# Repair RBT after Insert
📗 Repair a red root:
➩ If the root of the tree is red, switch it to black without violating any other property.
📗 Repair red node with red child:
➩ If parent's sibling (aunt) is black or null: (rotate) then rotate and color swap.
➩ If parent's sibling is red: recolor then check.
# RBT Insert Example
ID:📗 [1 point] Insert the following nodes: (click the "Insert" square to insert this node to the BST).
➩ Rotate the tree from A to B by dragging A to B in the diagram.
📗 List of nodes:
# RBT Insert Another Example
📗 Another example from Campus section: 7, 14, 18, 23, 1, 11, 20, 29, 25, 27
# Deleting from a RBT
📗 Delete value using the BST deletion algorithm.
📗 Check RBT properties and restore if necessary.
# Repair RBT after Delete
📗 If deleted node is black and replacement is red, turn replacement black.
📗 Otherwise:
➩ rotate and color swap then rotate and color swap and recolor.
➩ recolor then check.
➩ rotate and color swap then check.
# RBT Delete Example
ID:📗 [1 point] Delete the following nodes: (click the "Delete" square to delete this node from the BST).
➩ Rotate the tree from A to B by dragging A to B in the diagram and mark a node as a double by dragging the node to anywhere else.
📗 List of nodes:
📗 Use left child to replace:
# RBT Delete Same Example
📗 Drawing null nodes might be easier in some cases.
ID: 📗 [1 point] Delete the following nodes: (click the "Delete" square to delete this node from the BST).
➩ Rotate the tree from A to B by dragging A to B in the diagram and mark a node as a double by dragging the node to anywhere else.
📗 List of nodes:
📗 Use left child to replace:
# RBT Delete Another Example
📗 Another example from Campus section: 14, 7, 5, 1, 6, 11, 9, 13, 20, 18, 15, 19, 23, 29 (RED: 5, 11, 15, 19, 29)
📗 Delete: 15, 23, 20, 13, 29, 1, 5, 6
# Complexity of RBT Operations
📗 Search: \(O\left(H\right) = O\left(\log\left(N\right)\right)\)
📗 Insert: \(O\left(H\right) = O\left(\log\left(N\right)\right)\)
📗 Delete: \(O\left(H\right) = O\left(\log\left(N\right)\right)\)
# Testing
📗 3 dimensions for tests:
➩ Unit test ----- Integration test
➩ Opaque-box test ----- Clear-box test
➩ Input-output test ----- Property test
# Readings
📗 Notes from the Campus section: Link
📗 JAR file guide: Link
📗 JUnit assertions class: Link
📗 JUnit user guide: Link
test rbt,rin,rde,rdn l
📗 Notes and code adapted from the course taught by Professor Florian Heimerl and Ashley Samuelson.
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📗 You can expand all the examples and demos: , or print the notes: .
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Last Updated: November 25, 2025 at 1:46 AM