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Exploring Path Lengths and Tree Structures in Binary Trees

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This lecture covers essential concepts in binary trees (BTs), including the relationship between internal paths length (IPL) and external paths length (EPL). We will prove the relationship EPL(ni) = IPL(ni) + 2*ni using induction, exploring examples and tree construction. Key topics include the unique identification of trees from traversal sequences and essential tree class methods in object-oriented programming. Prepare for Exam 1 with upcoming review questions and project results, and note that it will be held on Monday, the 16th.

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Exploring Path Lengths and Tree Structures in Binary Trees

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Presentation Transcript

  1. CMSC 341Lecture 9

  2. Announcements • Proj 1 results • Next week • Exam 1 Mon 16 • review questions will appear on website

  3. Path Lengths • There is a relationship between the IPL and EPL of full BTs. If ni is the number of internal nodes in a FBT, then • EPL(ni) = IPL(ni) + 2ni • Example: • ni = • EPL(ni) = • IPL(ni) = • 2 ni =

  4. Proof of Path Lengths • Prove: EPL(ni) = IPL(ni) + 2 ni by induction • Base: ni = 0 (single node, the root) • EPL(ni) = 0 • IPL(ni) = 0; 2 ni = 0 0 = 0 + 0 • Assume for N: True for all FBT with ni < N • Prove for N+1: Let niL, niR be # of int. nodes in L, R subtrees. • niL + niR = N - 1 ==> niL < N; niR < N • EPL(niL) = IPL(niL) + 2 niL • EPL (niR) = IPL(niR) + 2 niR • EPL(ni) = EPL(niL) + EPL(niR) + niL + 1 + niR + 1 • IPL(ni) = IPL(niL) + IPL(niR) + niL + niR • EPL(ni) = IPL(ni) + 2 ni

  5. Constructing Trees • Is it possible to reconstruct a BT from just one of its pre-order, inorder, or post-order sequences?

  6. Constructing Trees (cont) • Given two sequences (say pre-order and inorder) is the tree unique?

  7. Tree Implementations • What should methods of a tree class be?

  8. Tree class • template <class Object> • class Tree { • public: • Tree(const Object &notFnd); • Tree (const Tree &rhs); • ~Tree(); • const Object &find(const Object &x) const; • bool isEmpty() const; • void printTree() const; • void makeEmpty(); • void insert (const Object &x); • void remove (const Object &x); • const Tree &operator=(const Tree &rhs);

  9. Tree class (cont) • private: • TreeNode<Object> *root; • const Object ITEM_NOT_FOUND; • const Object &elementAt(TreeNode<Object> *t) const; • void insert (const Object &x, TreeNode<Object> * &t) const; • void remove (const Object &x, TreeNode<Object> * &t) const; • TreeNode<Object> *find(const Object &x, • TreeNode<Object> *t) const; • void makeEmpty(TreeNode<Object> *&t) const; • void printTree(TreeNode<Object *t) const; • TreeNode<Object> *clone(TreeNode<Object> *t)const; • };

  10. Tree Implementations • Fixed Binary • element • left pointer • right pointer • Fixed K-ary • element • array of K child pointers • Linked Sibling/Child • element • firstChild pointer • nextSibling pointer

  11. TreeNode : Static Binary • template <class Object> • class BinaryNode { • Object element; • BinaryNode *left; • BinaryNode *right; • BinaryNode(const Object &theElement, BinaryNode *lt, BinaryNode *rt) : element (theElement), left(lt), right(rt) {} • friend class Tree<Object>; • };

  12. TreeNode : Static K-ary • template <class Object> • class KaryNode { • Object element; • KaryNode *children[MAX_CHILDREN]; • KaryNode(const Object &theElement; • friend class Tree<Object>; • };

  13. TreeNode : Sibling/Child • template <class Object> • class KTreeNode { • Object element; • KTreeNode *nextSibling; • KTreeNode *firstChild; • KTreeNode(const Object &theElement, KTreeNode *ns, KTreeNode *fc) : element (theElement), nextSibling(ns), firstChild(fc) {} • friend class Tree<Object>; • };

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