In nested hierarchies, the species branching off into their own clades become increasingly reproductively isolated. At first it's because they're geographically isolated from each other or that they have become sexually isolated such that they don't recognize each other as potential mates. The potential for hybrids is there early after the branching off, but eventually they become too different genetically so that they are physically unable to produce offspring. At that point, you simply cannot have clades merging into each other.
This is a rather idealised view of how life works - in many ways it's much more similar to language that you are implying here.
Hybridisation is not just a possibility early after the branching off (though of course you may be using 'early' different to how I would). Populations of animals have a tendency to anastomose - they split off from one another and become distinct due to geographic isolation, but upon being brought back into contact readily hybridise. This happens among populations whose shared ancestry is believed to be very distant; and which are often classified as different genera or even families (the most absurd example a quick internet search could turn up was between echinoderms classed in different orders; with estimated divergences in the Triassic or even Permian!).
These hybrids aren't necessarily odd dead ends, either - there's evidence of consistent gene flow from baboons into geladas, for example, despite the fact that these populations remain distinct (reminiscent of loanwords in language).
To complicate the tree thing further, reproductive isolation is not something that evolves consistently. There's no guarantee that two seperated populations will develop any genetic incompatibility over a specified time span, which can lead to an odd situation known from some ostariophysan fish. Population A splits from population B. Population B splits into B1 and B2. B1 then evolves some genetic incompatibility with the others so they cannot produce fertile hybrids. B2, then, can still freely interbreed with A should they come back into contact, but not with the more closely related B2.
This is all just animals, but of course with other organisms the tree picture is even fuzzier. Plants can hybridise across far greater distances in some circumstances - humans have exploited this a lot in making cultivars. And plants themselves, of course, contain the descendants of at least two distinct endosymbiotic bacteria within their cells - some of whose DNA has been incorporated into the plant genome. It gets even more complicated in other eukaryotes, who have endosymbiotic eukaryotes, which have themselves endosymbiotic eukaryotes. I'm not sure I explained that well, but what I mean is that there's a eukaryotic algae, which is incorporated within another organism, which is itself incorporated within another organism. This means that the parent of all this endosymbiosis gradually winds up incorporated the DNA of 5 different organisms in its genome.
Except that's not all! This tertiary symbiote also has all sorts of other DNA incorporated by endoviruses, some of which have themselves carried DNA from still other organisms. Transfer of genes among bacteria is common, and is what prompted the infamous and misleading 'Darwin was wrong' cover on New Scientist. I remember reading the suggestion somewhere that distinctive layers in bacterial mats, traditionally classified as different taxonomic groups, were in reality the result of persistent gene transfer between bacteria in close contact; which natural selection sorting the most appropriate genes into the different layers. Don't know if this was just idle speculation or not, though.