I am going to attempt to make an espalier fence ( image1) with my grafted apple trees that I planted about month back.  I have no experience what so ever doing this other then what I am reading about on the internet.  I am pretty sure I can do this but I have a question that I can not find the answer to so I am going to see if any here has attempted this or has any knowledge on this subject.

I have 6 different varieties of apple trees already planted in the ground that are 11 feet apart. What would happen if in the years to come when the branches on each row came close enough to connect I decided to graft them together ? ?  This would then so to say link all 6 trees together side by side all the way down the line ... Would there be any benefit to connect them together and would there be a downside to connecting together ? ? ? I will attempt to make a drawing of exactly what I am talking about so you may get a better idea...  I am not an artist so please bare with my drawing

I will be making the espalier fence with cables running from one end to the other so that I can train the trees along the same wire.  Image 2 is my ( so called Pablo Picasso drawing ) where I was thinking about joining them together.

Fascinating! I don't know nearly enough about grafting trees to give an answer, but I'm really curious to see if anyone with more experience knows. I know traits from one tree transfers to another via grafting (which is why rootstock can give disease resistance and stature traits to the tree grafted to it). I also wonder how you'd graft them together. Would you graft them into a Y or just straight into each other, so they each dead-end into each other?

I attached my horrible rendition of trees getting grafted.

If you Google "tree shaping" you'll find all sorts of pictures and articles about how to bend and fuse trees to become a single entity. 

Here's an article from the UK Permaculture Magazine that talks about it.

https://www.permaculture.co.uk/articles/artful-science-tree-shaping

So, yes, it's possible to do what you are proposing.  Some trees like willows and sycamores take to being shaped and fused much easier than other varieties. 

Check out these images to see the vast range of possibilities.

https://www.google.com/search?q=tree+shaping&safe=off&tbm=isch&tbo=u&source=univ&sa=X&ved=0ahUKEwiKkZCTgLjVAhUrrFQKHRhZAqIQsAQIMQ&biw=1276&bih=680

I have never done it but it is possible. It is done with pleached rows of trees like tillia (lime) and hornbeam here and makes for a stronger structure and think the trees would share nutrients and water to some degree too. Maybe the downsides would be the extra work and the potential of disease transmission in certain situations.

The mention of diseases is germane.  Any latent viruses present in one of the joined trees will spread into the other, and also into the rootstock of the other.  There is a host of viruses and viroids normally present in apples, most of which never cause any problems.  But some of them do cause adverse effects in some cultivars, but absolutely none in others.  Since you generally have no idea of the virus status of your trees, (unless you are growing more modern trees with virus-indexed rootstock and scion wood).  So, it will probably be very satisfactory.  But there is a small possibility of causing probelms wity virus transmission.  (For some neat ideas as to just how far one can push the idea of grafting multiple trees together, look at http://newatlas.com/full-grown-tree-furniture-munro/37709/. ; Munro grafts branches of a single tree back into themselves to form solid unions between individual branches.  But the same concept works identically with two or more individual trees grafted together.

Yes you can join trees as long as they are from the same family (apples in this case), as David mentions, there can be draw backs but these are not usual.

Living fences have been joined for centuries and many of the old ones are still thriving after a few hundred years.

To join them all you really need to do is expose the cambium layer where the branches cross and then tie them together, use some Elmer's white glue to coat the joint to prevent insects and disease out and you are good to go.
You can use rubber band material to tie them and that will rot away around the time the wounds heal over. If you use cordage then you will want to keep an eye on the wraps so that you can change them as the branches grow together so there isn't any deformation.

Redhawk

Years ago I saw a tunnel like arbor made entirely out of fused pear trees. The trees were 8' on center  and arched up to about 11' high where they curved over and were grafted to the tree on the opposite side and the branches were trained laterally and grafted together. The tunnel was filled with pears and cooler than outside the structure.

Jason

One thing You might consider.  If You graft the trees together, rather than prune them as separate intensities, then as one tree gets a disease; it might spread to all trees grafted together.  Where separate trees can be treated separately.

If you look up some of the Bonsai techniques you will see various techniques for grafting  two separate plants of the same species , most are done by removing bark carefuly
Doing it as they grow from young plants that are already established  is one way .

You are best to decide how far apart your " fence posts "  will be , then plant the young trees that far  apart . Put your post mid way in between the saplings  & run the straining wires between the posts without too much tension on them .

Start to train a " leader branch " in spring wern things are supple & full of sap . Make them  lightly &  softly fastened to the the wire guide to the adjacent trunk at the height that they grow or gently use the straining wires as anchors to secure a cord tied lightly to a branch thats at a slightly different height .  It's up to you how you do it , two low leaders off one trunk supplying a tree on either side or only taking one leader off each trunk & taking it to just one sapling  at each level .

One interesting way is to use a slow speed battery drill or a sharp hand borer and bore a hole with a wood boring twist drill an inch or so into the trunk the diameter of the leader that you have decided to grow where it will enter the new trunk .

You usually take it off an adjacent tree using grafting wax & self amalgamating rubber tape to seal the hole & keep the leader in place in the new trunk .
 
A demonstration I saw was where a long leader from the top of a young tree it was gently slowly but surley  has been pulled right down in a letter " P " by hand then pulled through the hole that went right through the trunk , sealed with grafting wax , braced with 5 mm dia Bonsai soft wire that had bee run through some soft plastic tube to stop it cutting in at the points of contact to secure it and left to grow .
This was done to make two new low down branches to balance th tree in later years .  It can also be used to put two lateral leaders in low down .

Again where the leader passes through the drilled trunk you will need to remove outer & the inner bark in the prescribed manner so you have only exposed the  Cambian layer showing .  Don't over do it and cut too much out so you only have  wood left ...they won't grow .

One side of these cleaned of bark ( At the point of joining only )  leader must be a good contact fit to the cambium layer of the host tree so use a caliper to measure the diameter to discover what drill size you will need .

It's worthwhile spending a few hours or so learning how to cut the outer bark off then shaving the inner bark to understand what the cambium layer looks like on each particular tree.

If you go too far and cut everything out till you only have the wood of the tree , there  will not be any path of nutrients to pass through & the cutting / graft will fail

Most of these sort of graftings are done in early spring when the sap starts to rise in the host plants as the growing season then gives you a far greater chance of the host pant accepting the newcomer .



Re disease on your tree fence ..
If there is any disease that will totally kill a single tree off in  a single attack  then yes it could be a problem ..happily there are not many such diseases that cannot be treated using natural products that fully comply with the sites ethos .

As you'll have many trees close together most of the time any other disease can be cut out and the bare wound sealed with a  simple sealing compound .

I see Banyan trees here that have naturally grafted together - just by having tons of roots and branches rubbing against each other for a long time, they eventually get fused. This is two parts of one tree getting grafted together, a little different than your idea but it occurs naturally and its pretty cool.

I created a "Belgiam Fence" using different varieties of apple trees. It is still growing great, decades later. One issue it has is that some of the limbs/varieties grew at different rates than others, so there are thicker and thinner branches in the weave. Looks unsymmetrical. I'm still damned proud of it.  I planted it partially as a car-catcher: To stop cars that might be flying off the road towards the house. It is incredibly strong for that purpose.

If we want to speculate that the trees might share viruses between them, then we may also want to speculate that they might share immune responses, so that if any tree in the clump is resistant, that they all are.