Gareth Williams-Gardner, on whose ground the tree stands, and neighbour David Evans were each fined £1,500 and ordered to pay £400 in costs and victim surcharge at Poole Magistrates Court last week, after pleading guilty to wilful damage of the mature cedar tree in the town's Twemlow Avenue in July.
The two claimed debris had fallen from the tree, posing a risk to life and property.
The borough's senior arboriculturist Andy Osborne said: "This was a significant tree making a positive contribution to the street scene, and the unauthorised works have diminished its value within the public realm."
Cabinet portfolio holder for planning and regeneration Ian Potter added: "This conviction clearly demonstrates the council’s commitment to protecting Poole’s environment.
"The council will have no hesitation in seeking prosecution through the courts against those individuals who seek to damage or destroy trees."