Well, there's inspiration in the methods used by villagers in the article.
Apparently the villagers grow the bridges by guiding them through pipes:
In order to make a rubber tree's roots grow in the right direction--say, over a river--the Khasis use betel nut trunks, sliced down the middle and hollowed out, to create root-guidance systems. The thin, tender roots of the rubber tree, prevented from fanning out by the betel nut trunks, grow straight out. When they reach the other side of the river, they're allowed to take root in the soil. Given enough time, a sturdy, living bridge is produced.
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Elves may not want to chop out wood, but I could see elves using cloth or plaster to do the same thing. This would require rope/cloth construction, which has a precedent with the rope ladders people have been talking about since forever.
So say you have a Hometrees growing a number of "buds" or branches around their trunk. If you want to build a bridge between two such trees, you can build a rope bridge - quick, but fragile and only one tile wide. On the other hand you can build a guide-rope for the same purpose, and the two branches will grow together over a period of years, which can be sped up by having some kind of tree-singer or druid do their thing. Eventually you have a 1-tile bridge that thickens every few years until it's large enough to build houses on.