6-Parenthesis and Binary Trees - Parenthesis and Binary...

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Parenthesis and Binary Trees Consider the 4 node binary tree: 1 / \ 2 4 / \ / \ 3 / \ If traverse the tree in preorder then the numbers assigned to the nodes reflect the order in which they are visited. Now, suppose create a sequence of 1’s and 0’s by traversing the tree in preorder and writing down a 1 when we visit a node and a 0 when we encounter a null subtree; except for the last null subtree . For this tree we get the sequence 1 1 0 1 0 0 1 0 Now, suppose we substitute “(“for each 1 and “)” for each 0 of the sequence. We get: ( ( ) ( ) ) ( ) - a valid sequence of parentheses. In general, we can represent any n node binary tree uniquely by such a sequence of n1’s and n 0’s or n left and n right parentheses. Now, following Knuth, let A pq be a lexicographic listing of all strings α composed of p left and q > p right parentheses for which the string, ( q-p α, is a valid sequence of parentheses, where 0 < p < q and q >0. We take A 00 to be the null string and A pq to be the null list when p<0 or q<p. Then, the first string in the listing A pq must be ) q-p ( ) . . . ( ) where there are p of the pairs ( ) and the last string in the listing must be ( p ) q . Why? Can describe the listing A pq as) Apq -1 + (A p-1q where )A pq means every string of A pq is preceded by ) and similarly for (A pq . Why is this recursive description correct?
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  • Fall '16
  • James Korsh
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