I've stumbled a heap-like sorting algorithm on the Internet as followed:$\\$For convenience, given a list of $2^n \; (n \in \mathbb{N^*})$ distinct numbers to be sorted increasingly.Step 1: From the list, build a $P$-tree (to be defined) satisfying the min-heap property (Willard Zhan points out that this is actually a binomial heap), which costs $2^n-1$ comparisons. Go to Step 2.Step 2: The root of the tree is the minimum on the tree at this time. So, pop it out and have $T$ trees remained. If $T > 1$, go to Step 3. If $T = 1$, go to Step 2. Otherwise, stop.Step 3: By comparing their roots, join 2 of the remaining trees into 1 tree. The joining is done as followed: the larger root will be another child of the smaller root. The choice of 2 trees is done as followed: choose 2 trees with the least numbers of root's children. Continue joining until there is only 1 tree remained. One joining costs $1$ comparison, the whole step costs $T - 1$ comparisons ($T$ is taken from the previous step). Go to Step 2.$P$-trees are defined as followed: $P_0$ is the $1$-node tree. For all positive integer $m$, $P_m$ is the tree having exactly $m$ first-class subtrees $P_0, ..., P_{m-1}$.$\\$I don't know this algorithm's proper name (if any). So I can't find any analysis on it, especially on its number of comparisons. I post this Question to ask for help on its name, its analysis, or some sources to read about them.Thanks in advance.
Analysis on comparisons in a heap-like sorting algorithm
algorithms;sorting;trees;heaps;heap sort