Hmm. Strange advice, and illogical. If only half the tree is felled the first year, the rest will still be very much alive and, as is the way with Eucalypts, will sprout profusely from the wounds to continue growing vigorously. It will not be half dead, which is what I think the thinking is here. Indeed, if it is then felled to ground level, it will still be alive  Eucalpytus are famous for regenerating from the base following fire in their natural habitat. The stump would need to be treated to kill it.

Further, I can't think that the roots will potentially damage the nearby foundations as they decompose. They will only do this if they have found their way into brick work or under footings in the first place. Do you know if they have? If there is any damage, you'll know about it and it isn't going to get any worse by having the roots die. It's if they keep growing that any problems are likely to continue. But Eucalyptus are deep rooted, tending not to go sideways like those usual foundation-of-building-damaging Willows and Poplars. So I don't suppose there's going to be a problem.

Sensible advice in many cases, actually, though Eucs are (usually) so small-crowned, that I doubt it will matter here.

The reason for doing so is to reduce the tree's water demand by stages; take half off now, and the water content in the soil will rise, but not by as much as if the tree were taken down and the roots treated with herbicide in one go. If the water content in the soil rises sharply, it can result in heave damage to the foundations, though usually only if the tree is much older than the house (i.e., the house was built on already-dehydrated, sunken ground), which is very unlikely in this case (given that a 10 metre Euc is probably only about 7 or 8 years old).

Simon - sorry, what you say is wrong: willows and poplars are very deep-rooted, which is precisely why they are so damaging. Their roots go deep enough to get underneath the foundations, and dry and shrink the soil below, leaving the foundations unsupported and liable to subsidence.

Shallow-rooted trees like birch and beech are the ones least likely to cause subsidence damage to foundations.

Resin