The Tree Data Model. Laura Kovács
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1 The Tree Data Model Laura Kovács
2 Trees (Baumstrukturen) Definition Trees are sets of points, called nodes (Knoten) and lines, called edges (Kanten), connecting two distinct nodes, such that: n 2 n 3 n 4 there is one special node, called the root (Wurzel); Ex: every node c other than the root is connected by an edge to some other node p. Node p is called the parent (Vorgänger) of node c; Ex: n 2 is parent of n 5, n 6 Node c is called the child (Nachfolger) of node p; Ex: n 5, n 6 are children of n 2 the tree is connected, that is: if we start at any node n different than the root move to the parent of n move to the parent of parent of n... reach the root of the tree. Ex: n 7 n 4
3 Trees (Baumstrukturen) Definition Trees are sets of points, called nodes (Knoten) and lines, called edges (Kanten), connecting two distinct nodes, such that: n 2 n 3 n 4 there is one special node, called the root (Wurzel); Ex: every node c other than the root is connected by an edge to some other node p. Node p is called the parent (Vorgänger) of node c; Ex: n 2 is parent of n 5, n 6 Node c is called the child (Nachfolger) of node p; Ex: n 5, n 6 are children of n 2 the tree is connected, that is: if we start at any node n different than the root move to the parent of n move to the parent of parent of n... reach the root of the tree. Ex: n 7 n 4
4 Trees (Baumstrukturen) Definition Trees are sets of n 2 n 3 n 4 points, called nodes (Knoten) and lines, called edges (Kanten), connecting two distinct nodes, such that: there is one special node, called the root (Wurzel); Ex: every node c other than the root is connected by an edge to some other node p. Node p is called the parent (Vater) of node c; Ex: n 2 is parent of n 5, n 6 Node c is called the child (Sohn) of node p; Ex: n 5, n 6 are children of n 2 the tree is connected, that is: if we start at any node n different than the root move to the parent of n move to the parent of parent of n... reach the root of the tree. Ex: n 7 n 4
5 Trees (Baumstrukturen) Definition Trees are sets of n 2 n 3 n 4 points, called nodes (Knoten) and lines, called edges (Kanten), connecting two distinct nodes, such that: there is one special node, called the root (Wurzel); Ex: every node c other than the root is connected by an edge to some other node p. Node p is called the parent (Vater) of node c; Ex: n 2 is parent of n 5, n 6 Node c is called the child (Sohn) of node p; Ex: n 5, n 6 are children of n 2 the tree is connected, that is: if we start at any node n different than the root move to the parent of n move to the parent of parent of n... reach the root of the tree. Ex: n 7 n 4
6 Trees (Baumstrukturen) Definition Trees are sets of n 2 n 3 n 4 points, called nodes (Knoten) and lines, called edges (Kanten), connecting two distinct nodes, such that: there is one special node, called the root (Wurzel); Ex: every node c other than the root is connected by an edge to some other node p. Node p is called the parent (Vater) of node c; Ex: n 2 is parent of n 5, n 6 Node c is called the child (Sohn) of node p; Ex: n 5, n 6 are children of n 2 the tree is connected, that is: if we start at any node n different than the root move to the parent of n move to the parent of parent of n... reach the root of the tree. Ex: n 7 n 4 A node with no children is called a leaf (Blatt). Ex: n 5, n 6, n 7 are leaves.
7 Trees (Baumstrukturen) Alternative Definition A single node n is a tree. n is said to be the root of this tree.
8 Trees (Baumstrukturen) Alternative Definition A single node n is a tree. n is said to be the root of this tree. Let r be a new node, and T 1,, T k trees with roots c 1,,c k. Then a new tree T can be formed by - make r the root of T; - add an edge from r to each c 1,, c k. r c 1 c 2 c k T 1 T 2 T k
9 Trees (Baumstrukturen) Alternative Definition A single node n is a tree. n is said to be the root of this tree. Let r be a new node, and T 1,, T k trees with roots c 1,,c k. Then a new tree T can be formed by - make r the root of T; - add an edge from r to each c 1,, c k. r Trees T 1,, T k are subtrees (Teilbäume) of r. c 1 c 2 c k Note: T i contains c i ; the root of T i is c i. T 1 T 2 T k Note: A subtree with root c contains all the children of c, the children of children of c, etc.
10 Trees (Baumstrukturen) Alternative Definition A single node n is a tree. n is said to be the root of this tree. n 2 n 3 n 4 Let r be a new node, and T 1,, T k trees with roots c 1,,c k. Then a new tree T can be formed by - make r the root of T; - add an edge from r to each c 1,, c k. r Trees T 1,, T k are subtrees (Teilbäume) of r. Ex: is a subtree. is not a subtree. n 2 n 2 c 1 c 2 c k T 1 T 2 T k n 5 n 6 n 5 Note: A subtree with root c contains all the children of c, the children of children of c, etc.
11 - m k-1 is the parent of m k. Ex:, n 2, n 6 is a path of length 2. Trees Path A path (Pfad) in a tree is a sequence of nodes m 1,m 2,m 3,,m k such that: - m 2 is the parent of m 1, n 2 n 3 n 4 - m 3 is the parent of m 2, Note: (m 1,m 2 ), (m 2,m 3 ),,(m k-1,m k ) are edges of the tree. Between arbitrary two nodes there is exactly one path. The length (Länge) of the path is k-1. Ex: is a path of length 0.
12 - m k-1 is the parent of m k. Ex:, n 2, n 6 is a path of length 2. Trees Path A path (Pfad) in a tree is a sequence of nodes m 1,m 2,m 3,,m k such that: - m 2 is the parent of m 1, n 2 n 3 n 4 - m 3 is the parent of m 2, Note: (m 1,m 2 ), (m 2,m 3 ),,(m k-1,m k ) are edges of the tree. Between arbitrary two nodes there is exactly one path. The length (Länge) of the path is k-1. Ex: is a path of length 0.
13 Trees Path A path (Pfad) in a tree is a sequence of nodes m 1,m 2,m 3,,m k such that: - m 2 is the parent of m 1, - m 3 is the parent of m 2, - m k-1 is the parent of m k. Ex:, n 2, n 6 is a path of length 2. Ex: is a path of length 0. Note: (m 1,m 2 ), (m 2,m 3 ),,(m k-1,m k ) are edges of the tree. Between arbitrary two nodes there is exactly one path. The length (Länge) of the path is k-1. n 2 n 3 n 4 m 1 is called an ancestor (Vorgänger) of m k ; m k is a descendant (Nachfolger) of m 1. Ex: is an ancestor of n 2, n 6 ; n 6, n 2 are descendants of.
14 Trees Height, Depth, Degree n 2 n 3 n 4 The height (Höhe) of node m is the length of the longest path from m to a leaf. Ex: Height of is 2, height of n 2 is 1, leaf n 5 has height 0. The height of a tree is the height of the root. Ex: Height of the tree is 2. The depth/level (level) of node m is the length of the path from the root to m. Ex: Depth of is 0, depth of n 2 is 1, leaf n 5 has depth 2. The degree (Ordnung) of a tree is the maximum of the number of subtrees of nodes. Ex: Degree of the tree is 3.
15 Trees Height, Depth, Degree Level 0 n 2 n 3 n 4 Level 1 Level 2 Height of the tree is 2. Degree of the tree is 3.
16 Trees Ordered Trees (geordnete Baum) n 2 n 3 n 4 An ordered tree (geordneten Baum) is a tree where an order is assigned to the children of any node. Example: Assign a left-to-right order to the children of any node. Then, among the children of : n 2 is the leftmost child of, then n 3, then n 4. -n 4 is the rightmost child of. -n 3 is to the left of n 4. In an ordered tree (geordneten Baum) the order of the subtrees is relevant.
17 Trees Isomorphic Trees Trees who differ only by the order of their subtrees are isomorphic. Example of isomorphic trees: n 3 n 2 n 4 n 4 n 3 n 2 n 2 n 3 n 4 n 7 n 5 n 6
18 Trees Binary Trees (binärer Baum) A binary tree is a tree such that each node has maximum two subtrees. Special binary tree: empty tree (no nodes, no edges). Note: The degree of a binary tree is maximum 2. Ex: n 2 n 3 n 4 n 2 n 4 is not a binary tree is a binary tree Binary trees have left (linken) and right (rechten) subtrees.
19 Difference between a Tree and a Binary Trees BINARY TREE A binary tree may be empty. No node in a binary tree may have more than 2 subtrees. Degree of a binary tree is maximum 2. Subtrees of a binary tree are ordered. TREE A tree cannot be empty. No limit on the number of subtrees of a node in a tree. No limit on the degree of a tree. Subtrees of a tree are not ordered.
20 Difference between a Tree and a Binary Trees BINARY TREE A binary tree may be empty. No node in a binary tree may have more than 2 subtrees. Degree of a binary trees is maximum 2. The subtrees of a binary tree are ordered. TREE A tree cannot be empty. No limit on the number of subtrees of a node in a tree. No limit on the degree of a tree. Subtrees of a tree are not ordered. Ex: a a - different when viewed as a binary tree b c c b - same when viewed as a tree
21 Full (Perfect/Complete) Binary Trees (perfekter/voll binärer Baum) A binary tree is full / complete / perfect when and the left subtree the right subtree of each node contains the same number of nodes. Ex: n 2 n 4 n 2 n 4 is not a full binary tree n 5 n 6 n 3 n 7 is a full binary tree
22 Full (Perfect/Complete) Binary Trees (perfekter/voll binärer Baum) A binary tree is full / complete / perfect when and the left subtree the right subtree of each node contains the same number of nodes. Ex: n 2 n 4 n 2 n 4 is not a full binary tree n 5 n 6 n 3 n 7 is a full binary tree In a full binary tree each node - is either a leaf; - or has exactly two non-empty subtrees.
23 Full Binary Trees In a full binary tree with N nodes and height h: and N = 2 h+1-1 h = ld(n+1)-1 A full binary tree with height h has exactly 2 h leaves.
24 Binary Trees - Syntax Trees (Syntaxbaum) Syntax tree (expression tree) is a binary tree of an arithmetic expression. Nodes: arithmetic operators (+,-,*, ) and numbers/variables Leafs: numbers/variables Edges: parent-child relation between nodes is defined by the precedence of operators (indicated by parentheses). * + c Example: (a+b)*c a b
25 Binary Trees - Syntax Trees (Syntaxbaum) 1. The syntax tree of operand a is a single-node tree with root labeled by a. a 2. If T 1 is the syntax tree of arithmetic expression A 1, and T 2 is the syntax tree of the arithmetic expression A 2, then: 2.1. the expression tree of A 1 op 2 A 2, where op 2 is a binary operator (+,*,-, ), is: op 2 Binary operator = operator with 2 arguments A 1 A the expression tree of op 1 A 1, where op 1 is a unary operator (!, ld, ), is: op 1 Unary operator = operator with 1 arguments A 1
26 Binary Trees - Syntax Trees (Syntaxbaum) Example: (a+b)*c Example: (a+b)*ld(c) * * + c + ld a b a b c
27 Binary Trees Traversal of Binary Trees Prefix traversal Infix traversal Postfix
28 Binary Trees Prefix Traversal PREFIX / PREORDER Traversal Prefix / Preorder notation (polish notation): Recursively perform the following operations: Visit the node; Traverse left subtree; Traverse right subtree. (Also called: depth-first traversal.) preorder(root) { print root.value; if NotEmpty(root.left) then preorder(root.left); if NotEmpty(root.right) then preorder(root.right) } Example: (a+b)*c + a b * c Prefix/Preorder notation: *+abc For a node n, let n.value denote its value, n.left its left subtree, n.right its right subtree.
29 Binary Trees Infix Traversal INFIX / INORDER Traversal Infix / Inorder notation: Recursively perform the following operations: Traverse the left subtree; Visit the node; Traverse the right subtree. inorder(root) { if NotEmpty(root.left) then inorder(root.left); print root.value; if NotEmpty(root.right) then inorder(root.right) } Example: (a+b)*c + a b * c Infix/Inorder notation: a+b*c For a node n, let n.value denote its value, n.left its left subtree, n.right its right subtree.
30 Binary Trees Postfix Traversal POSTFIX / POSTORDER Traversal Postfix / Postorder notation (reverse polish notation): Recursively perform the following operations: Traverse the left subtree; Traverse the right subtree; Visit the node. (Also called breadth-first traversal.) postorder(root) { if NotEmpty(root.left) then postorder(root.left); if NotEmpty(root.right) then postorder(root.right); print root.value } Example: (a+b)*c + a b * c Postfix/Postorder notation: ab+c* For a node n, let n.value denote its value, n.left its left subtree, n.right its right subtree.
31 Binary Trees Binary Search (Sort) Tree (Sortierbaum) A binary search tree is a binary tree with: The left subtree of a node n contains only nodes with values (keys) less than the value of n; The right subtree of a node n contains only nodes with values (keys) greater than the value of n; Both the left and right subtrees of n must be also binary search trees. Note: Each node has a distinct value. Inorder traversal of a binary search tress yields a sorted list of nodes. Example: d Inorder: abcde SORTED LIST of NODES Preorder: dbace Postorder: acbed Levelorder: d be ac (listing nodes from left-to-right, level-by-level starting from root) a b c e
32 Binary Trees Binary Search (Sort) Tree (Sortierbaum) A binary search tree is a binary tree with: The left subtree of a node n contains only nodes with values (keys) less than the value of n; The right subtree of a node n contains only nodes with values (keys) greater than the value of n; Both the left and right subtrees of n must be also binary search trees. Note: Each node has a distinct value. Inorder traversal of a binary search tress yields a sorted list of nodes. Let T be a binary search tree. Let Nodes(T) denote the set of nodes of T. For a node n of T, let: n.left denote its left subtree; n.right denote its right subtree; n.value denote the value of n. Then: "n: nœnodes(t): (" n l : n l œnodes(n.left): n l.value<n.value) (" n r : n r œnodes(n.right): n r.value>n.value) An alternative: "n: nœnodes(t): (" n l : n l œnodes(n.left): n l.valuebn.value) (" n r : n r œnodes(n.right): n r.value>n.key)
33 Binary Trees - Exercises Consider the expression a b + c d - * e f + / in postfix form. What is its infix form? What is its prefix form? Consider the binary tree: 8 Is it a binary search tree? Is it a full binary tree? What is the degree of the tree? What is the height of the tree? What is its prefix form?
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