Handout 4.0 - TREES.pdf

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TREES A is the root node B is the parent to D& E A is the ancestor/grand parent of D& E D and E are the descendants of A C is the sibling of B. G and E are not siblings H,I,J,K,L,M,N,O are leaves. Leaves have no children. A node which is not a leaf is known as an internal node. A,B,C,D,E,F,G are internal nodes A is at level zero. The depth (level) of H is 3. …………………….level 0 ………….. level 1 ……level 2 ……level 3 The height of the tree is the maximum levels the tree has. In the above case, 3 The degree of node is the number of nodes it has. All the nodes above each have a degree of 2. The tree structure represent a hierarchy e.g. the organization structure of a corporation , a basic family unit table of content of a book , the file system of an operation system. An ordered tree is one in which the children of each node has utmost two nodes. Like the tree above. Binary tree An example of binary trees- arithmetic expression (((2+3)4)+(4*3)/8)-(12+(7-(3+2))) O A B D H I E J K L M F C N G O A B D H I E J K L M F C N G - / + 4 + 2 3 4 3 x 8 12 + - 7 + 2 3
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2 The tree above is complete because: At the i th level, there are 2 i nodes – it’s full. Every internal node having two children is not necessarily right to say it is complete. Check arithmetic example above If h is the height of the tree (for the tree above, height = 3). It means : - All the leaves are at level h. - The number of levels in the tree is 2 h - The number internal nodes is given by = 1 + 2 + 2 2 + … + 2 h-1 = 2 h – 1 - Total number of nodes is 2 h + 2 h – 1 = 2 h +1 – 1 = n If the tree is complete and has n nodes, - The number of leaves is (n + 1)/2 - The height of the tree is = log 2 (number of leaves) ADTs for trees generic container method : o size(), isEmpty(), element() Positional container method : o positional( ) - gives all position in trees as a sequence. It has no parameters, when invoked in a tree, it gives a sequence of all the nodes in the tree. o swapElement(p,q) – swap the elements at this positions . o replaceElement(p,e) – replaces the element at the position, p with e Query methods: o isRoot (p) – given a particular position , is the root of the tree? o IsInternal (P) – given a particular position in the tree, is P on an internal node o isExternal (p) – given a particular position , does it correspond to a leaf accessor method o root() – will return the position of the root o parent(p) – given a particular position P, the returns the position of the parent node O A B D H I E J K L M F C N G
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3 o children(p) – given a particular position, p, return the children of the node. The children (if they are many) will be returned as a sequence of the positions.
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