Difficult - if the trees are mature, planting directly beneath them won't be possible, and any plant too close to the roots will starve or suffer from drought. What are the trees?

Clematis montana,virginia creeper,boston ivy,russian vine,golden hop,honeysuckle etc ,all fast growing commonly stocked climbers.All would need a good planting pocket of soil/compost.

The Clematis will not grow up a tree trunk and need moist soil, which it wont get under a tree. I agree with Bamboo difficult it the trees are mature. Have you considered planting shrubs a metre in front of the trees, these would screen the trunks and have more access to light and moisture. What are these trees with 'tall unsightly trunks'?

rambling roses but as previoulsy said the root ball woud need to be 1m away from the tree trunks. Ivy if you like it but may need 'tying' in.

Hello Mmugford ... when you said tree trunks, are they live or dead trunks? I figure you would have said hedge trees if alive? If dead, then Dgw above has some good ideas. If this is the right track, then some trellis between or in front of the trunks will help to train quick growing climbers. Depending on where you are in the UK, passion flower is a very fast growing climber that you will need to keep tying in until it covers the area you need. Dgw's other suggestions are also good, including the clematis. If the trunks are 'live', then any climber you put in will need plenty of regular watering and feeding, and stones at its base to retain cool, moist temperatures at the base of the clematis, which hates its roots to get too warm. Hope this helps?!