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Trees

Trees. Definitions 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. Nodes Edges Root node Interior node Leaf Parent Children. Ancestor / Proper ancestor

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Trees

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  1. Trees • Definitions • 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

  2. 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 Definitions

  3. descendants of “a” 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

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

  5. Implementing a Tree A: 0 1 2 3 4 5 6 7 8 9 - A B C - - D - - - • One array • 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 ▲ ▲

  6. A B C ▲ ▲ ▲ D ▲ ▲ Implementing a Tree • Three array ▲=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 / / / / /

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

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

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

  10. 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

  11. 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

  12. K-ary Tree => Binary Tree a 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

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

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