Hi Guest are you saying the trees are in your garden? If so they belong to you not your neighbour. Very little will grow happily near big trees they take all the nourishment and water from the soil leaving it pretty impoverished. You could try making raised beds and planting in those. Without knowing which aspect the bed has and whether it is in a windy position it is not really possible to advise on specific shrubs.

You've told us there are large trees next door, but you haven't said whether these make your garden shady or not, nor how close the tree roots are to your garden. If the shrubs you'd already planted have died, are you saying the trees were the cause, or is it more likely you don't water when you need to, which will be necessary at times during dry spells, because the trees suck out all the moisture that's available. Other thing to ask is whether those trees are evergreen or not.