Complexity of union-find with path-compression, without rank
Seidel and Sharir proved in 2005 [1] that using path compression with arbitrary linking roughly on $m$ operations has a complexity of roughly $O((m+n)\log(n))$.

See [1], Section 3 (Arbitrary Linking):
Let $f(m,n)$ denote the runtime of union-find with $m$ operations and $n$ elements. They proved the following:


  Claim 3.1. For any integer $k&gt;1$ we have $f(m, n)\leq (m+(k−1)n)\lceil \log_k(n) \rceil$.


According to [1], setting $k = \lceil m/n \rceil + 1$ gives
$$f(m, n)\leq (2m+n) \log_{\lceil m/n\rceil +1}n$$.

A similar bound was given using a more complex method by Tarjan and van Leeuwen in [2], Section 3:


  Lemma 7 of [2]. Suppose $m \geq n$. In any sequence of set operations implemented using any form of compaction and naive linking, the total number of nodes on find paths is at most $(4m + n) \lceil \log_{\lfloor 1 + m/n \rfloor}n \rceil$ With halving and naive linking, the total number of nodes on find paths is at most $ (8m+2n)\lceil \log_{\lfloor 1 + m/n \rfloor} (n) \rceil $.
  
  Lemma 9 of [2]. Suppose $m &lt; n$. In any sequence of set operations implemented using compression and naive linking, the total number of nodes on find paths is at most $ n + 2m \lceil \log n\rceil + m$. 


[1]: R. Seidel and M. Sharir. Top-Down Analysis of Path Compression. Siam J. Computing, 2005, Vol. 34, No. 3, pp. 515-525.

[2]: R. Tarjan and J. van Leeuwen. Worst-case Analysis of Set Union Algorithms. J. ACM, Vol. 31, No. 2, April 1984, pp. 245-281.