For life forms that procreate by individuals splitting themselves into two copies, while also getting DNA from each other in various random or intentional ways, it gets a lot messier. We still talk about "species", but it's a quite different thing.
The closest to merging is probably mitochondria, which probably were single-cell creatures ingested by some very early (single-celled) ancestor of us and, instead of being devoured, ended up in a symbiotic relationship within its host. The mitochondria to this day have their own genome and function a lot like a cell-within-a-cell. Same for plants and photosynthesis.
It’s important to note that such viral infections generally do not infect germline cells and are thus not automatically inherited (though, like in the case of genital herpes, are easily transmitted to the offspring during birth).
That’s a second point, and I should have corrected the start (to “three-ish, since the point on mitochondria is yet another)
If you’re asking specifically about instances of non-treelike gene transfer, the answer is twofold:
First, and I may be mistaken here (I’ve been out of the field for a few years) I think that any metric fulfilling the triangle inequality has a corresponding, consistent tree. So you’ll get a tree, totally. It may just not be the correct one.
Second, if you are asking for our ability to quantify divergence from that model, I’ll give a short explanation of the codon usage method mentioned before as an example:
The genetic code has 4 letters (ACGT) that translates to a different sequence consisting of 21 Amino Acids (a chain of amino acids is a protein, such as the enzymes doing catalysing all the fancy chemical reactions in your body)
Each three letters of DNA code one “letter” of the protein sequence, meaning one amino acid. Since 4^3 > 21, the code isn’t quite optimal.
There are some three-letter sequences that contain meta instructions like start/stop, and some that don’t have any meaning. But there are also instances of two or more three-letter sequences translating to the same amino acid. They are functionally identical. You can replace them in the lab, without any change in phenotype.
For some reason, some species (or even branches in the teee of life) still show preferences for using one or the other three-letter code for a given amino acid.
If you plot which of the (functionally identical) codes are used within a bacterial genome, you can find sudden changes, where the preference switches dramatically for some lenght, then returns to the previous preference.
That’s indicative of a piece of DMA that jumped across the tree. It’s really not very subtle when you know to look for it.
Another, even more obvious, example is viral DNA: this tends to end up as mangled, non-coding (“junk”) DNA, containing (fragments of) genes that often have nothing in common with the rest of the DNA, but have long stretches nearly identical to, say, a known gene for some viral coat.
In terms of quantification I’m really at the limits of my memory (and, unfortunately, in-flight internet), but I’ll take a stab and the former mechanism can be found in 5-10% of bacterial genomes, and amount to usually less than 8% of he genome, with maybe one or two exceptional cases with 20% or so (some bacterial species may have developed a tendency to exploit this sort of buffer overflow to cheat at evolution)
Phylogenists (the biologists working in trees, but not those “trees”) do consider all this (and much more). Where they encounter “known unknowns”, you will often see trees with nodes branching into more than two branches. That’s essentially what cartographers would lable “here be dragons”. Or, prosaically: it isn’t quite clear which split happened first in evolutionary history.
Have we done the same sort of experiment with genetic code to verify whether it actually does form a tree, or if that's just an assumption we are forcing on the data?
The only confounding factor is that among bacteria, there is a process called “horizontal gene transfer”, where one bacterium inadvertently (?) aquires some genetic information from another when they’re really just out for a quick snack. Plus viruses, which sometimes just get inserted into a host’s genome and becomes part of it, sometimes even functional. The way animals acquired mitochondria is also a fascinating divergence from the standard “tree” model, although it’s somewhat derivative of how plants acquired photosynthesis.
(That’s how everything in biology works. It’s the worst case of spaghetti coding, ever, full of self-modifying compilers, state saved in the JIT’s buffer, and programs only working by exploiting CPU flaws depending on the Hall effect, but only on workdays (according to the Julian calendar), fan speed, and your database server’s support of OpenGL).
But it’s really just a tree.
What’s far less of a “real” concept, is that of species. That makes essentially no sense in non-sexually reproducing organisms.