Definitions of complete trees are typically limited to some specific kind of tree, often an $m$-ary tree, where the number of children each internal node must have is a positive integer $m$.
Consider the situation where $m$ is varied as a function of the level $L$ of a rooted tree such that each internal node on level $L$ must have exactly $f(L) = m_L^{}$ children. Because there is no constant value of $m$, the tree is not strictly considered an $m$-ary tree.
Given such a tree, determining how many nodes exist on each level is trivial; simply take the product of each $m_L$ value from $L=0$ to some terminal value. It would also be simple to construct an algorithm that is able to determine whether all possible nodes in such a tree have been assigned a value within the constraints of traditional definitions of completeness in trees; the only thing I'm concerned with is that traditional definitions that I've seen tend to limit themselves to one type of tree or another.
My question is this: if I can prove that such a "variable $m$-ary tree" has no empty nodes, am I able to designate such a tree as complete without having to provide modifications to traditional definitions of complete trees? In other words, is there something essential in traditional definitions of complete trees that I am missing?