# CSCI Trees. Mark Redekopp David Kempe

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1 1 CSCI Trees Mark Redekopp David Kempe

2 2 Properties, Insertion and Removal BINARY SEARCH TREES

3 3 Binary Search Tree Binary search tree = binary tree where all nodes meet the property that: All values of nodes in left subtree are less-than or equal than the parent s value All values of nodes in right subtree are greater-than or equal than the parent s value If we wanted to print the values in sorted order would you use an pre-order, in-order, or post-order traversal?

4 4 BST Insertion Important: To be efficient (useful) we need to keep the binary search tree balanced Practice: Build a BST from the data values below To insert an item walk the tree (go left if value is less than node, right if greater than node) until you find an empty location, at which point you insert the new value Insertion Order: 25, 18, 47, 7,, 32, 56 Insertion Order: 7, 18,, 25, 32, 47, 56

5 5 BST Insertion Important: To be efficient (useful) we need to keep the binary search tree balanced Practice: Build a BST from the data values below To insert an item walk the tree (go left if value is less than node, right if greater than node) until you find an empty location, at which point you insert the new value Insertion Order: 25, 18, 47, 7,, 32, 56 Insertion Order: 7, 18,, 25, 32, 47, A major topic we will talk about is algorithms to keep a BST balanced as we do insertions/removals 47 56

6 6 Successors & Predecessors Let's take a quick tangent that will help us understand how to do BST Removal Given a node in a BST Its predecessor is defined as the next smallest value in the tree Its successor is defined as the next biggest value in the tree Where would you expect to find a node's successor? Where would find a node's predecessor? m

7 7 Predecessors If left child exists, predecessor is the right most node of the left subtree Else walk up the ancestor chain until you traverse the first right child pointer (find the first node who is a right child of his parent that parent is the predecessor) If you get to the root w/o finding a node who is a right child, there is no predecessor 10 Pred(50) Pred(25)

8 8 Predecessors If left child exists, predecessor is the right most node of the left subtree Else walk up the ancestor chain until you traverse the first right child pointer (find the first node who is a right child of his parent that parent is the predecessor) If you get to the root w/o finding a node who is a right child, there is no predecessor Pred(50) = Pred(25)=

9 9 Successors If right child exists, successor is the left most node of the right subtree Else walk up the ancestor chain until you traverse the first left child pointer (find the first node who is a left child of his parent that parent is the successor) If you get to the root w/o finding a node who is a left child, there is no successor 10 Succ() Succ()

10 10 Successors If right child exists, successor is the left most node of the right subtree Else walk up the ancestor chain until you traverse the first left child pointer (find the first node who is a left child of his parent that parent is the successor) If you get to the root w/o finding a node who is a left child, there is no successor 10 Succ() = Succ()=

11 11 BST Removal To remove a value from a BST First find the value to remove by walking the tree If the value is in a leaf node, simply remove that leaf node If the value is in a non-leaf node, swap the value with its in-order successor or predecessor and then remove the value A non-leaf node's successor or predecessor is guaranteed to be a leaf node (which we can remove) or have 1 child which can be promoted We can maintain the BST properties by putting a value's successor or predecessor in its place Remove 25 Remove Remove Either or 25 Swap w/ pred Leaf node so just delete it 25 1-Child so just promote child 25 is a non-leaf so can't delete it where it is swap w/ successor or predecessor 10 Swap w/ succ

12 12 Worst Case BST Efficiency Insertion Balanced: Unbalanced: Removal Balanced: Unbalanced: Find/Search Balanced: Unbalanced: #include<iostream> using namespace std; // Bin. Search Tree template <typename T> class BST { public: BTree(); ~BTree(); virtual bool empty() = 0; virtual void insert(const T& v) = 0; virtual void remove(const T& v) = 0; virtual T* find(const T& v) = 0; };

13 13 BST Efficiency Insertion Balanced: O(log n) Unbalanced: O(n) Removal Balanced : O(log n) Unbalanced: O(n) Find/Search Balanced : O(log n) Unbalanced: O(n) #include<iostream> using namespace std; // Bin. Search Tree template <typename T> class BST { public: BTree(); ~BTree(); virtual bool empty() = 0; virtual void insert(const T& v) = 0; virtual void remove(const T& v) = 0; virtual T* find(const T& v) = 0; };

14 14 Trees & Maps/Sets C++ STL "maps" and "sets" use binary search trees internally to store their keys (and values) that can grow or contract as needed This allows O(log n) time to find/check membership BUT ONLY if we keep the tree balanced! Map::find("Greg") key value Map::find("Mark") Returns iterator to corresponding pair<string, Student> "Jordan" Student object Returns iterator to end() [i.e. NULL] "Frank" Student object "Percy" Student object "Anne" Student object "Greg" Student object "Tommy" Student object

15 15 An example of B-Trees 2-3 TREES

16 16 Definition 2-3 Tree is a tree where Non-leaf nodes have 1 value & 2 children or 2 values and 3 children All leaves are at the same level Following the line of reasoning All leaves at the same level with internal nodes having at least 2 children implies a (full / complete) tree FULL (Recall complete just means the lower level is filled left to right but not full) A full tree with n nodes implies Height that is bounded by log 2 (n) 0 1 a 2 Node 4 a 3 Node 2 4 a valid 2-3 tree

17 17 Implementation of 2- & 3-Nodes You will see that at different times 2 nodes may have to be upgraded to 3 nodes To model these nodes we plan for the worst case a 3 node This requires wasted storage for 2 nodes template <typename T> struct Item23 { T val1; T val2; Item23<T>* left; Item23<T>* mid; Item23<T>* right; bool twonode; }; a 2 Node _ a 3 Node

18 Search Trees Similar properties as a BST 2-3 Search Tree If a 2 Node with value, m Left subtree nodes are < node value Right subtree nodes are > node value If a 3 Node with value, l and r Left subtree nodes are < l Middle subtree > l and < r Right subtree nodes are > r 2-3 Trees are almost always used as search trees, so from now on if we say 2-3 tree we mean 2-3 search tree < m a 2 Node m > m m = "median" or "middle" < l a 3 Node l > l && < r l = left r = right r > r

19 Search Tree Binary search tree compared to 2-3 tree Check if 55 is in the tree? BST Tree

20 2-3 Insertion Algorithm Key: Since all leaves must be at the same level ("leaves always have their feet on the ground"), insertion causes the tree to "grow upward" To insert a value, 1. walk the tree to a leaf using your search approach 2a. If the leaf is a 2-node (i.e.1 value), add the new value to that node 2b. Else break the 3-node into two 2-nodes with the smallest value as the left, biggest as the right, and median value promoted to the parent with smallest and biggest node added as children of the parent Repeat step 2(a or b) for the parent Insert 60,, 10,, 25, 50, 80 Key: Any time a node accumulates 3 values, split it into single valued nodes (i.e. 2-nodes) and promote the median Empty Add 60 Add Add 10 Add

21 Insertion Algorithm Key: Since all leaves must be at the same level ("leaves always have their feet on the ground"), insertion causes the tree to "grow upward" To insert a value, 1. walk the tree to a leaf using your search approach 2a. If the leaf is a 2-node (i.e.1 value), add the new value to that node 2b. Else break the 3-node into two 2-nodes with the smallest value as the left, biggest as the right, and median value promoted to the parent with smallest and biggest node added as children of the parent Repeat step 2(a or b) for the parent Insert 60,, 10,, 25, 50, 80 Key: Any time a node accumulates 3 values, split it into single valued nodes (i.e. 2-nodes) and promote the median Add 25 Add

22 Insertion Algorithm Key: Since all leaves must be at the same level ("leaves always have their feet on the ground"), insertion causes the tree to "grow upward" To insert a value, 1. walk the tree to a leaf using your search approach 2a. If the leaf is a 2-node (i.e.1 value), add the new value to that node 2b. Else break the 3-node into two 2 nodes with the smallest value as the left, biggest as the right, and median value promoted to the parent with smallest and biggest node added as children of the parent Repeat step 2(a or b) for the parent Insert 60,, 10,, 25, 50, 80 Key: Any time a node accumulates 3 values, split it into single valued nodes (i.e. 2-nodes) and promote the median Add

23 Insertion Algorithm Key: Since all leaves must be at the same level ("leaves always have their feet on the ground"), insertion causes the tree to "grow upward" To insert a value, 1. walk the tree to a leaf using your search approach 2a. If the leaf is a 2-node (i.e.1 value), add the new value to that node 2b. Else break the 3-node into two 2 nodes with the smallest value as the left, biggest as the right, and median value promoted to the parent with smallest and biggest node added as children of the parent Repeat step 2(a or b) for the parent Insert 90,91,92, 93 Add

24 Insertion Algorithm Key: Since all leaves must be at the same level, insertion causes the tree to "grow upward" To insert a value, 1. walk the tree to a leaf using your search approach 2a. If the leaf is a 2-node (i.e.1 value), add the new value to that node 2b. Else break the 3-node into two 2 nodes with the smallest value as the left, biggest as the right, and median value should be promoted to the parent with smallest and biggest node added as children of the parent Repeat step 2(a or b) for the parent Insert 90,91,92,93 Add 90 Add

25 25 Add Insertion Algorithm Key: Since all leaves must be at the same level, insertion causes the tree to "grow upward" To insert a value, 1. walk the tree to a leaf using your search approach 2a. If the leaf is a 2-node (i.e.1 value), add the new value to that node 2b. Else break the 3-node into two 2 nodes with the smallest value as the left, biggest as the right, and median value should be promoted to the parent with smallest and biggest node added as children of the parent Repeat step 2(a or b) for the parent Insert 90,91,92,

26 26 Add Insertion Algorithm Key: Since all leaves must be at the same level, insertion causes the tree to "grow upward" To insert a value, 1. walk the tree to a leaf using your search approach 2a. If the leaf is a 2-node (i.e.1 value), add the new value to that node 2b. Else break the 3-node into two 2 nodes with the smallest value as the left, biggest as the right, and median value should be promoted to the parent with smallest and biggest node added as children of the parent Repeat step 2(a or b) for the parent Insert 90,91,92,

27 Tree Removal Key: 2-3 Trees must remain "full" (leaf nodes all at the same level) Remove 1. Find data item to remove 2. If data item is not in a leaf node, find in-order successor (which is in a leaf node) and swap values (it's safe to put successor in your location) 3. Remove item from the leaf node 4. If leaf node is now empty, call fixtree(leafnode) fixtree(n) If n is root, delete root and return Let p be the parent of n If a sibling of n has two items Redistribute items between n, sibling, and p and move any appropriate child from sibling to n Else Choose a sibling, s, of n and bring an item from p into s redistributing any children of n to s Remove node n If parent is empty, fixtree(p) Another key: Want to get item to remove down to a leaf and then work up the tree

28 28 P L S L - S P Remove Cases Redistribute 1 Redistribute 2 Merge 1 Merge 2 L P P L S L - S P a b c d a b c d - S S L S - S P a b c a b c P = parent S = smaller L = larger Empty root S - L S L a b c a b c

29 29 Remove Remove Examples Key: Keep all your feet (leaves) on the ground (on the bottom row) Remove Not a leaf node so swap w/ successor at leaf Can't just delete because a 3- node would have only 2 children Since 2 items at leaf, just remove Rotate 60 down into 50 to make a 3-node at the leaf and 2-node parent

30 P L S L - S P Remove Cases Redistribute 1 Redistribute 2 Merge 1 Merge 2 P P P L S L - S P a b c d a b c d - S S P S - S P a b c a b c Empty root - S L S L a b c a b c

31 31 Remove 80 Remove Examples Rotate parent down and empty node up, then recurse 90 Internal so swap w/ successor at leaf - Remove root and thus height of tree decreases Rotate parent down and empty node up, then recurse

32 32 P L S L - S P Remove Cases Redistribute 1 Redistribute 2 Merge 1 Merge 2 P P P L S L - S P a b c d a b c d - S S P S - S P a b c a b c Empty root - S L S L a b c a b c

33 33 Remove Remove Exercise Step 1: Not a leaf, so swap with successor Step 2: Remove item from node Step 3: Two values and 3 nodes, so merge. Must maintain levels.

34 34 Remove Exercise 1 (cont.) 50 Start over with the empty parent. Do another merge Step 4: Merge values Step 5: Can delete the empty root node

35 35 Remove 50 Remove Exercise Step 1: It s a leaf node, so no need to find successor. Remove the item from node. Step 2: Since no 3- node children, push a value of parent into a child. 90 Step 3: Delete the node

36 36 P L S L - S P Remove Cases Redistribute 1 Redistribute 2 Merge 1 Merge 2 P P P L S L - S P a b c d a b c d - S S P S - S P a b c a b c Empty root - S L S L a b c a b c

37 37 Insert 12 Insertion Exercise

38 38 Insert Insertion Exercise

39 39 Insert 39 Insertion Exercise

40 40 Removal Exercise 4 Remove

41 41 Removal Exercise 5 Remove

42 42 Remove Remove Exercise

43 43 Other Resources /BTree.html

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