Trees Definitions Implementation Traversals K-ary Trees Read Weiss, 4.1 – 4.2 Implementation Nodes and Links One Arrays Three Arrays Traversals Preorder, Inorder, Postorder K-ary Trees Converting trees: k-ary ↔ binary

Definitions Nodes Edges Root node Interior node Leaf Parent Children Ancestor / Proper ancestor Descendant / Proper descendant Level of a node Height of a node / tree Degree of a node / tree Depth of node Path Acyclic graph

a j b k c g m d l i h f e descendants of “a” root, height=4, depth=level=0 degree=1 proper descendants of “a” j b k c interior node, height=2, depth=level=2 degree=2 g m d l i h f e degree=0 leaf, height=0, depth=level=4 degree of tree = 2 height of tree = 4

Implementing a Tree Nodes and Links A B C D Node { Object value; Node lchild; Node rchild; } // Node A B C ▲ ▲ ▲ D ▲=null link ▲ ▲

Implementing a Tree One array A B C D A: 0 1 2 3 4 5 6 7 8 9 A[1] is root lchild of A[i] is A[2i] rchild of A[i] is A[2i+1] “-” means array element is null / not used A[0] not used as a node A[0] may be used to hold general info (e.g., number of nodes in tree) A B C ▲ ▲ ▲ D ▲=null link ▲ ▲

Implementing a Tree Three array A B C D ▲ ▲=null link root = 4 freelist = 7 null = “/” = -1 0 1 2 3 4 5 6 7 8 9 Object value: D - - - A - B - C - int lchild: / / 3 5 6 1 / 9 0 2 int rchild: / / / / 8 / / / / /

Traversals Preorder N L R preorder (Node t) if (t == null) return; visit (t.value()); preorder (t.lchild()); preorder (t.rchild()); } // preorder

Traversals Inorder L N R inorder (Node t) if (t == null) return; inorder (t.lchild()); visit (t.value()); inorder (t.rchild()); } // inorder

Traversals Postorder L R N postorder (Node t) if (t == null) return; postorder (t.lchild()); postorder (t.rchild()); visit (t.value()); } // postorder

a j b k c g m d l i h f e preorder: a j k m l b c g i h d f e inorder: m k l j a b i g h c f d e postorder: m l k j i h g f e d c b a

K-ary Trees a q c e f b d n m i p g k j degree of tree = 4 degree of nodes f and n = 3 height of tree = 3 depth=level of m = 2

K-ary Tree => Binary Tree q c e f b d n m i p K-ary Binary root root leftmost child left child right sibling right child g k j

Traversals K-ary Tree Binary Tree n g e j k m d c b f i p q a n g e j Preorder: Inorder: Postorder: Preorder: Inorder: Postorder: