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A 2-way merge, or a binary merge, has been studied extensively due to its key role in merge sort. An example of such is the classic merge that appears frequently in merge sort examples. The classic merge outputs the data item with the lowest key at each step; given some sorted lists, it produces a sorted list containing all the elements in any ...
If we have a pointer to a node x in a Min HBLT, we can delete it as follows: Replace the node x with the result of merging its two subtrees and update the s-values of the nodes on the path from x to the root, swapping the right and left subtrees if necessary to maintain the leftist tree property.
In a binary tree that represents a multi-way tree T, each node corresponds to a node in T and has two pointers: one to the node's first child, and one to its next sibling in T. The children of a node thus form a singly-linked list. To find a node n 's k 'th child, one needs to traverse this list:
To merge the two trees, apply a merge algorithm to the right spine of the left tree and the left spine of the right tree, replacing these two paths in two trees by a single path that contains the same nodes. In the merged path, the successor in the sorted order of each node from the left tree is placed in its right child, and the successor of ...
Because each binomial tree in a binomial heap corresponds to a bit in the binary representation of its size, there is an analogy between the merging of two heaps and the binary addition of the sizes of the two heaps, from right-to-left. Whenever a carry occurs during addition, this corresponds to a merging of two binomial trees during the merge.
Join follows the right spine of t 1 until a node c which is balanced with t 2. At this point a new node with left child c, root k and right child t 2 is created to replace c. The new node may invalidate the balancing invariant. This can be fixed with rotations. The following is the join algorithms on different balancing schemes.
For example, merging heaps is done simply by concatenating the two lists of trees, and operation decrease key sometimes cuts a node from its parent and forms a new tree. However, at some point order needs to be introduced to the heap to achieve the desired running time.
Initially when each node is the root of its own tree, it's trivially true. Assume that a node u with rank r has at least 2 r nodes. Then when two trees with rank r are merged using the operation Union by Rank, a tree with rank r + 1 results, the root of which has at least + = + nodes.