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Trees

Trees. Briana B. Morrison Adapted from Alan Eugenio. Topics. Trees Binary Trees Definitions (full, complete, etc.) Expression Trees Traversals Implementation. Tree Structures. Computers”R”Us. Sales. Manufacturing. R&D. US. International. Laptops. Desktops. Europe. Asia. Canada.

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Trees

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  1. Trees Briana B. Morrison Adapted from Alan Eugenio

  2. Topics • Trees • Binary Trees • Definitions (full, complete, etc.) • Expression Trees • Traversals • Implementation Binary Trees

  3. Tree Structures Binary Trees

  4. Computers”R”Us Sales Manufacturing R&D US International Laptops Desktops Europe Asia Canada What is a Tree • In computer science, a tree is an abstract model of a hierarchical structure • A tree consists of nodes with a parent-child relation • Applications: • Organization charts • File systems • Programming environments Binary Trees

  5. Binary Trees

  6. Terminology • First let’s learn some terminology about trees… Binary Trees

  7. Binary Trees

  8. Tree Structures Binary Trees

  9. Binary Trees

  10. Tree Structures Binary Trees

  11. Binary Trees

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  14. Tree Structures Binary Trees

  15. Binary Trees

  16. Binary Trees

  17. 50, 80, 90, 84 Binary Trees

  18. BINARY TREES Binary Trees

  19. Binary Tree Definition • A binary tree T is a finite set of nodes with one of the following properties: • (a) T is a tree if the set of nodes is empty. (An empty tree is a tree.) • (b) The set consists of a root, R, and exactly two distinct binary trees, the left subtree TL and the right subtreeTR. The nodes in T consist of node R and all the nodes in TL and TR. Binary Trees

  20. Binary Trees

  21. Binary Tree • Applications: • arithmetic expressions • decision processes • searching • A binary tree is a tree with the following properties: • Each internal node has two children • The children of a node are an ordered pair • We call the children of an internal node left child and right child • Alternative recursive definition: a binary tree is either • a tree consisting of a single node, or • a tree whose root has an ordered pair of children, each of which is a binary tree A C B D E F G I H Binary Trees

  22. Examples of Binary Trees • Expression tree • Non-leaf (internal) nodes contain operators • Leaf nodes contain operands • Huffman tree • Represents Huffman codes for characters appearing in a file or stream • Huffman code may use different numbers of bits to encode different characters • ASCII or Unicode uses a fixed number of bits for each character Binary Trees

  23. Examples of Huffman Tree Code for b = 100000 Code for w = 110001 Code for s = 0011 Code for e = 010 Binary Trees

  24. Binary Trees

  25. HOW CAN WE DEFINE leaves(t), THE NUMBER OF LEAVES IN THE BINARY TREE t? RECURSIVELY! Binary Trees

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  33. height(t) = ? To distinguish the height of an empty tree from the height of a single-item tree, What should the height of an empty tree be? Binary Trees

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  37. Tree Node Level and Path Length Binary Trees

  38. Depth Example Depth 0 (root) Depth 1 Depth 2 Depth 1 Binary Trees

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  49. Tree Node Level and Path Length – Depth Discussion Is this tree complete? Is it full? Binary Trees

  50. Binary Trees

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