trench

The root trench comes from Old French trenchier, "to cut, to slice," which gave English both a cutting tool and a cut in the ground. The oldest English sense is the most physical: a trench is literally a long cut made in the earth — a ditch dug for drainage, for planting, or for war.

The military meaning dominates the family. To dig a trench is to cut a protective channel you can crouch in while the enemy fires overhead. World War I made the trenches a byword for grinding, dug-in warfare. And that physical act of "digging in" is where the figurative members are born.

entrench is en- (to put into) + trench: to place something inside a trench, to dig it in. A soldier entrenched behind earthworks is hard to dislodge — and that is exactly the metaphor. Today entrenched almost always means figurative: an entrenched habit, entrenched corruption, entrenched interests are all "dug in" so deep you can't easily root them out. The image of a soldier who refuses to leave his ditch sits underneath every modern use.

retrench takes a different turn from the same cut. re- (back) + trench originally meant to cut something back, to pare it down — and in modern English it means to cut back spending: a company forced to retrench lays off staff and trims budgets. retrenchment is the noun: a period of cost-cutting and shrinking. Here the "cut" is financial, not military, but it is the same blade.

So the family splits cleanly by prefix. en- puts you into the trench (dig in, become fixed and immovable). re- cuts things back (reduce, economize). Both keep the original idea of trenchier: making a decisive cut.