In a disjoint-set forest, the representative of each set is the root of that set's tree . " Find " follows parent nodes until it reaches the root . " Union " combines two trees into one by attaching the root of one to the root of the other.
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This siftUp version can be visualized as starting with an empty heap and successively inserting elements, whereas the siftDown version given above treats the entire input array as a full but " broken " heap and " repairs " it starting from the last non-trivial sub-heap ( that is, the last parent node ).
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This can be interpreted as mapping the infinite depth binary tree onto this tree and then applying breadth-first search : replace the " down " edges connecting a parent node to its second and later children with " right " edges from the first child to the second child, from the second child to the third child, etc . Thus at each step one can either go down ( append a (, 1 ) to the end ) or go right ( add one to the last number ) ( except the root, which is extra and can only go down ), which shows the correspondence between the infinite binary tree and the above numbering; the sum of the entries ( minus one ) corresponds to the distance from the root, which agrees with the 2 " n " " 1 nodes at depth in the infinite binary tree ( 2 corresponds to binary ).
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