如何验证给定树是否为Binomial Tree（二项式树）？

First, let's anchor ourselves to the strict definition of a binomial tree—this is non-negotiable for validation:

• A rank-0 binomial tree (B₀) is just a single node with no subtrees.
• A rank-r binomial tree (Bᵣ) is formed by merging two rank-(r-1) binomial trees: you make the root of one tree the leftmost child of the other's root. This means the root of Bᵣ has exactly r subtrees, and those subtrees must be ordered by rank from r-1 down to 0, each being a valid binomial tree themselves.

Now, given your node structure ((rank, element, binomial_tree_list)), here's how to flesh out that recursive validation function you're thinking of:

Core Recursive Validation Logic #

Let's build a function is_valid_binomial_tree(tree) with these clear steps:

1. Base Case (Rank = 0)
   • Check if the tree's rank is 0.
   • Check if binomial_tree_list is empty.
   • If both conditions are true, it's a valid B₀.
2. Recursive Case (Rank = r > 0)
   • First, verify that the length of binomial_tree_list equals r—since a rank-r tree must have exactly r subtrees.
   • Next, validate each subtree in the list:
     • Their ranks must strictly decrease from r-1 down to 0 (first subtree is rank r-1, next r-2, ..., last is 0).
     • Each subtree must pass the is_valid_binomial_tree check recursively.
   • Only if all these conditions are met is the tree a valid Bᵣ.

Example Walkthrough #

Let's test a rank-2 tree to make this concrete:

• Its binomial_tree_list must have exactly 2 subtrees.
• The first subtree needs to be a valid rank-1 tree (which itself has 1 valid rank-0 subtree).
• The second subtree must be a valid rank-0 tree.
• If all these layers check out, we've confirmed it's a legitimate B₂.

Python-Style Pseudocode #

def is_valid_binomial_tree(tree):
    rank, element, subtrees = tree
    # Handle rank 0 case
    if rank == 0:
        return len(subtrees) == 0
    # Handle rank > 0 case: first check subtree count matches rank
    if len(subtrees) != rank:
        return False
    # Check each subtree's rank and validity
    expected_rank = rank - 1
    for subtree in subtrees:
        subtree_rank = subtree[0]
        if subtree_rank != expected_rank:
            return False
        if not is_valid_binomial_tree(subtree):
            return False
        expected_rank -= 1
    return True

This logic directly enforces all the key properties of binomial trees. If you're using this as part of a binomial heap (where elements have ordering constraints), you can add extra checks here (like verifying parent elements are >= children, depending on your heap type).

内容的提问来源于stack exchange，提问作者Adam Ralphus