Is cutting a branch from someone's house, which happened many years ago and in childhood, considered haram (forbidden)?

It is forbidden to transgress against a Muslim's property, whether by destruction, usurpation, or theft, due to the Almighty's saying: ﴿Do not consume your property among yourselves in falsehood﴾, and the Prophet's (peace and blessings be upon him) saying: «The property of a Muslim person is not lawful except with his willing consent.»

Whoever destroys a Muslim's property must guarantee it, whether the destroyer is minor or adult. He should pay its equivalent if it has an equivalent, or its value if it does not. So, if the branch you cut has value, you must pay it to its owner. If it has no value, then apologizing to him is sufficient.