Trees. Introduction & Terminology. February 05, 2018 Cinda Heeren / Geoffrey Tien 1

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1 Trees Introduction & Terminology Cinda Heeren / Geoffrey Tien 1

2 Review: linked lists Linked lists are constructed out of nodes, consisting of a data element a pointer to another node Lists are constructed as chains of such nodes List Cinda Heeren / Geoffrey Tien 2

3 Trees Root Trees are also constructed from nodes Nodes may now have pointers to one or more other nodes A set of nodes with a single starting point called the root of the tree (root node) Each node is connected by an edge to another node A tree is a connected graph There is a path to every node in the tree A tree has one less edge than the number of nodes Cinda Heeren / Geoffrey Tien 3

4 Is it a tree? Yes! No! Nodes are not all connected Yes! Although not a binary tree No! There is an extra edge (5 nodes, 5 edges) Yes! Actually equivalent to the blue tree Cinda Heeren / Geoffrey Tien 4

5 Tree relationships Node v is said to be a child of u, and u the parent of v if There is an edge between the nodes u and v, and edge root A Parent of B, C, D u is above v in the tree, This relationship can be generalized B C D E and F are descendants of A E F G D and A are ancestors of G B, C and D are siblings F and G are? Cinda Heeren / Geoffrey Tien 5

6 More tree terminology A leaf is a node with no children A path is a sequence of nodes v 1 v n where v i is a parent of v i+1 (1 i n) A subtree is any node in the tree along with all of its descendants A binary tree is a tree with at most two children per node The children are referred to as left and right We can also refer to left and right subtrees Cinda Heeren / Geoffrey Tien 6

7 Tree terminology example A A path from A to D to G A subtree rooted at B B C D E F G C, E, F, G are leaf nodes Cinda Heeren / Geoffrey Tien 7

8 Binary tree terminology A Left subtree of A B C Right child of A D E F G Right subtree of C H I J Cinda Heeren / Geoffrey Tien 8

9 Measuring trees The height of a node v is the length of the longest path from v to a leaf The height of the tree is the height of the root The depth of a node v is the length of the path from v to the root This is also referred to as the level of a node Note that there is a slightly different formulation of the height of a tree Where the height of a tree is said to be the number of different levels of nodes in the tree (including the root) Cinda Heeren / Geoffrey Tien 9

10 Tree measurements explained A Height of tree is 3 Height of node B is 2 B C Level 1 D E F G Level 2 Depth of node E is 2 H I J Level 3 Cinda Heeren / Geoffrey Tien 10

11 Perfect binary trees A binary tree is perfect, if No node has only one child And all the leaves have the same depth A perfect binary tree of height h has 2 h+1 1 nodes, of which 2 h are leaves Perfect trees are also complete Cinda Heeren / Geoffrey Tien 11

12 Height of a perfect tree Each level doubles the number of nodes Level 1 has 2 nodes (2 1 ) Level 2 has 4 nodes (2 2 ) or 2 times the number in Level 1 Therefore a tree with h levels has 2 h+1 1 nodes The root level has 1 node Bottom level has 2 h nodes, i.e. just over half of the nodes are leaves Cinda Heeren / Geoffrey Tien 12

13 Complete binary trees It's not quite perfect, but almost A binary tree is complete if The leaves are on at most two different levels, The second to bottom level is completely filled in and The leaves on the bottom level are as far to the left as possible D B E A F C Cinda Heeren / Geoffrey Tien 13

14 Balanced binary trees A binary tree is balanced if Leaves are all about the same distance from the root The exact specification varies Sometimes trees are balanced by comparing the height of nodes e.g. the height of a node s right subtree is at most one different from the height of its left subtree (e.g. AVL trees) Sometimes a tree's height is compared to the number of nodes e.g. red-black trees Cinda Heeren / Geoffrey Tien 14

15 "Balanced" binary trees A A B C B C D E F D E F G Cinda Heeren / Geoffrey Tien 15

16 "Imbalanced" binary trees A A B C B D E C D F Cinda Heeren / Geoffrey Tien 16

17 Binary tree traversal A traversal algorithm for a binary tree s each node in the tree Typically, it will do something while ing each node! Traversal algorithms are naturally recursive There are three traversal methods inorder preorder postorder Back to recursion Cinda Heeren / Geoffrey Tien 17

18 inorder traversal algorithm void inorder(node* nd) { if (nd!= NULL) { inorder(nd->leftchild); (nd); inorder(nd->rightchild); } } The function would do whatever the purpose of the traversal is (e.g. print the data value of the node). Cinda Heeren / Geoffrey Tien 18

19 preorder traversal (nd); preorder(nd->leftchild); preorder(nd->rightchild); preorder(left) preorder(right) 6 27 preorder(left) preorder(right) 3 9 preorder(left) preorder(right) preorder(left) preorder(right) preorder(left) 39 preorder(right) preorder(left) preorder(right) preorder(left) preorder(right) 11 Cinda Heeren / Geoffrey Tien 19

20 postorder traversal postorder(nd->leftchild); postorder(nd->rightchild); (nd); postorder(left) postorder(right) 7 27 postorder(left) postorder(right) 2 9 postorder(left) postorder(right) postorder(left) postorder(right) postorder(left) postorder(right) 5 6 postorder(left) 20 postorder(right) 39 postorder(left) postorder(right) Cinda Heeren / Geoffrey Tien 20

21 Exercise What will be printed by an in-order traversal of the tree? preorder? postorder? inorder(nd->leftchild); (nd); inorder(nd->rightchild); Food for thought: which traversal will be useful for deep copy? Deep delete? 39 Cinda Heeren / Geoffrey Tien 21

22 Readings for this lesson Carrano & Henry Chapter (Tree terminology, binary trees & traversals) Next class: Carrano & Henry: Chapter 15.3 (Binary search tree) Cinda Heeren / Geoffrey Tien 22

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