Question
I have a large mulberry tree in a row house backyard.
It splits into three trunks about 6ft off the ground and the canopy reaches into at least 5 of the backyards around mine.
Two of the splits lean right and left and one goes straight up.  I would like to trim all of the branches from the leaning trunks and leave the center.
Is this possible without killing the tree?  The trunks are each at least 2ft in diameter.

Answer
cutting the limbs off this large a tree is not recommended. It is not advisable to prune the trees heavily since the plant is inclined to bleed at the cuts. Cuts of more than two inches in diameter generally do not heal and should be avoided at all cost. The bleeding will be less severe if the tree is pruned while it is dormant.
What you need to do is think of giving the tree a "hair cut".
Proper pruning is more art than science. Keep in mind that if limbs need removing for safety (near roofs, blocking stop signs, etc.) or health (broken or diseased limbs) it should always be done and anytime of the year. For any pruning including the above or for height reduction or thinning out you should always keep these few rules in mind. If you follow these rules your completed job will look unpruned to the casual observer and the tree will be healthier.

Rule number 1: Never remove more than 1/3 of the tree's total leaf canopy in one year. If more pruning is needed then save some for next year. This is the most common mistake on younger trees but older trees also get butchered.

Rule number 2: Always remove a limb back to where its connected to its parent limb or trunk. Or cut it back to where another limb gowing from it.

This cut back to a side limb has the caveat that its diameter must be at least 1/3 the diameter of of the part removed. This rule alone should shoot holes in the poor practice of 'stub' pruning seen so frequently on mulberry trees. Following this rule prevents stub pruning from ever getting started. If, however your tree is stub pruned you can not correct and are stuck with annual butchering.

Rule number 3: Two thirds of the trees foliage should be lower than the rest. Trees in nature guide this rule. Rarely is seen a tree with most of the foliage in the to 1/3 of the tree nor do trees look like a large beach umbrella with a single trunk and flat spreading top. This is a common look after many people finish with trees. Most trees in nature form a silhouette pattern reminiscent of a mushroom's shape. This should be the model to guide pruning efforts.

Rule number 4: 80 to 90% of all wood removed should be done with 5 or fewer cuts. In other words a few selected cuts are much better than many small cuts. This demands more pre-cut planning and visualizing what will be left before making the cut.

And finally, Rule number 5: Topping is not acceptable pruning except for safety reasons as under power lines. If bids for pruning use the term 'topping', you'd be wise to get more bids. Topping is simple and usually results in a profusion of shoots rendering the tree more susceptible to wind damage. A tree with proper height reduction pruning will still maintain the look and shape of the tree, just shorter.

NOW all this maybe too complicate for the average homeowner so you may need to hire an arborist to prune the tree. Do not get just any tree trimmer. I would check with your local Agriculture Extension Service Horticultural expert and ask them for names of good arborist.