Conducting electricity is a matter simply of providing a medium through which electrons may pass and form a circuit. We use metal wires primarily because within a wire electrons flow so easily from one atom to another. Wires even have the wonderful property of conducting electric flow across the gap between two wires. This is handy, for example, when wiring up a new plug in the kitchen to run your new Whizzerizer 2000: so long as the two pieces of metal are sufficiently close, electrons can jump the gap and the juice will flow.
But other things than metal wires can conduct electricity. For example our bodies. Contrary to popular myth, our bodies are not made up "mostly of water". Closer to the truth, our bodies are made up of "mostly salt water". Water with stuff dissolved in it. The stuff dissolved in water creates an ionic solution -- positive & negative ions in free association within the watery medium -- and electricity can flow through that solution. This is why those "magic silver cleaning plates" you can buy on Amazon for only 19.99 work.
So it will be with your people's dendroupologiste. In this other world, trees live in a moist environment. Deep in the soil, their root tips stretch out into a medium of mineral matrix (soil) and water. Not so much free water as moisture bound up with the soil. What trees in this world also do, in addition to the usual chemo-fungal network communications trees ordinarily do, is have the added capacity to send electrical impulses via localised nodules of ionic solution. Basically, in the gap between two root tips, each tree exudes a tiny amount of salty water. This tiny droplet allows for electrical communication.
Those electrical impulses, generated deep within one tree's root body are carried through the fluid pathways down into the roots, leap the gap between root tips and travel up through the fluid pathways of another tree's roots and so on.
I leave for another question how your people can actually access this capacity of trees in order to use it as a functional computer!