Seeing a tree branch being cut in a dream symbolizes separation or divorce in a marriage context. The tree represents the source that sustains you while the branch or branches represents you or the person supported by the source. The branches can not survive on their own and hence needs a source to attach themselves to.
In a marriage context, the dream represents divorce. The tree is the husband who is the head of the family, while the branch represents the wife. A tree branch being cut in a dream means that the husband is separating himself from his wife or vice versa. We are only left with a tree without any branches and marriage is not complete without one partner. Hence, this dream represents divorce.
When you see a two-branched tree whose one branch is cut, this represents an end of polygamous marriage. The tree is the husband of two wives. The act of cutting one branch is a sign that the man will divorce one of his wives. I know a man who had two wives, and one of them had a similar dream. Eventually, this woman was cut off from the main trunk. She separated herself from the polygamous marriage system.
In other aspects of life, this dream may be a sign of separation or exclusion. This may be due to your lack of fruitfulness or unfaithfulness. If the tree represents the company you are working for, then it is a sign you will lose your job.
Related: Christian meaning of a door in a dream.
When the tree is a ministry, it is a sign of expulsion and separation due to lack of fruitfulness and unfaithfulness too. You are not adding value to the tree, rather you are just sucking its resources as a branch.
Jesus gave a similar example in John 15:2 by likening Himself to the tree.
He cuts off every branch in me that bears no fruit, while every branch that does bear fruit, he prunes so that it will be even more fruitful. John 15:2
Anyone who fails to add value to the kingdom of heaven will be cut off by God who is the vinedresser and Christ the vine. We believers in Jesus Christ are the branches.