Complex Relationships: Phylogenetic Tree

Complex Relationships: Language Tree

Other Examples of Trees

Files and folders on a computer Compilers: parse tree a = (b+c) * d; Genealogy trees These become complicated with some complex family relationships Lab 5: expression trees

First child/next sibling

class TreeNode { private: string element; TreeNode *firstChild; TreeNode *nextSibling; public: // ... }

Binary Trees

All nodes have at most 2 children class BinaryNode { public: // ... private: int element; BinaryNode *left; BinaryNode *right; };

Binary Trees: diagram details

In reality, any child not shown is really a NULL pointer, as shown here; but these are generally omitted from the diagrams

BST: insert

Do a find, and when we reach a NULL pointer, create a new node there

(no external source code)

void BST::insert(int x, BinaryNode * & curNode) {
    if (curNode==NULL)
        curNode = new BinaryNode(x,NULL,NULL);
    else if (x < curNode->element)
        insert(x, curNode->left);
    else if (x > curNode->element)
        insert(x, curNode->right);
    else
        ;    // duplicate... do nothing
}

BST: remove: no children

• Just remove the node (reclaiming memory), adjusting the parent pointer to NULL
  • In this case, 9's left child link is changed to NULL

→

BST: remove: one child

• Adjust pointer of parent to point at child, and reclaim memory
  • In this case, 4's left pointer is changed to point to 3

→

BST: remove: two children

• Replace node with successor, then remove successor from tree
  • This requires running findMin() on the right sub-tree, and then removing that element
  • In this case, 5 is replaced by 7 (and the node that had 7 is removed)

→

BST Height

n-node BST: Worst case depth is n-1 This can easily happen if the data to be inserted is already sorted Claim: The maximum number nodes in a binary tree of height h is 2h+1-1

AVL single right rotation

→

• The node just inserted was node 1 (blue)
• The lowest node, immediately after the insert, with an imbalance is node 3 (red)
• Because node 1 is in the "left subtree of the left child" of node 3, this means we need to perform a single right rotation

AVL single left rotation

→

• The node just inserted was node 3 (red)
• The lowest node, immediately after the insert, with an imbalance is node 1 (blue)
• Because node 3 is in the "right subtree of the right child" of node 1, this means we need to perform a single left rotation

A side-effect of tree rotations

→

• This is the single right rotation
• Note that at least one node moves "up" (depth decreases)
  • In this case, nodes 1 and 2 both move up
• And at least one node moves "down" (depth increases)
  • In this case, node 3 moves down
• Similarly for a left rotation

AVL single right rotation: before & after

→

• Node 1 (red) is what is being inserted
• The lowest node with an imbalance is node 5 (balance: -2)
• Because the insert was in 5's "left subtree of the left child", we perform a single right rotation on 5 (and its left child, 3)

AVL single right rotation: before & after

→

• From the previous slide, we know we perform a single right rotation on 5 (and its left child, 3)
• Thus, the two blue nodes are the 'pivots' of the rotation
• Note that node 4 changes parents (from 3's right to 5's left)

AVL single right rotation: general case

→

\( X < b < Y < a < Z \)

The insert is into sub-tree X, increasing its height to h+1

Notice how sub-tree Y changes parent

Cases 2 & 3: attempt a single rotation

→

\( X < b < Y < a < Z \)

The insert is into sub-tree Y, increasing its height to h+1

Failure! b's left subtree has height h+1; right is h+3

Double rotation

Node 5 (red) was just inserted The lowest node with an imbalance is node 8 (balance factor: -2) When discussing these rotations, we will call this the "parent" node Because the insert happened in 8's "right subtree of the left child", we perform a double rotation This consists of a single left rotation on the "child" (node 4), followed by a single right rotation on the "parent" (node 8)
Note that the two rotations are in different directions!

Double rotation, step 1

→

This is the single left rotation on the "child". The red node is what was inserted; the blue nodes are the 'pivots' of this single left rotation.

Double rotation, step 2

→

This is the single right rotation on the "parent". The red node is what was inserted; the green nodes are the 'pivots' of this single right rotation.

AVL double rotation: before & after

→

The red node is what was inserted

AVL double rotation: general case

→   \( W < b < X < c < Y < a < Z \)   The insert happens into X Notice sub-trees X and Y change parents

Algorithmic determination of rotation

left-left case	right-right case	left-right case	right-left case

• Given the lowest unbalanced node, and the child in the direction of the insert, compare the balance factors
• -2/+1 means a double left-right, +2/+1 means a singe left, etc.

All the tree rotations

The program tree