This could be caused by a fungal infection under the bark. It would help to know how old the tree is, its size and the growing conditions it has had for the past few years. From my limited knowledge fungal infections in any tree are hard to treat.

agree with Loosestrifes' questions - how long have you had the tree, how big is it, and where, precisely, is this split on the main trunk? Near the base, higher up? Has the split been weeping at all?

The tree is just over 3 years old - it is approx. 20ft. high
The split starts about 1 Ft above ground and is 1ft long.
It is about 3/4 mms deep. It is not weeping.
The tree has previously had 3 'accidents' with major growth being broken off [by neighbours and wind] these wounds have all healed up ok. and new shoots have grown from below them.
There is a large root now on the surface of the ground next to the tree - that I have not seen in previous years. [I don't know if this is from this tree or another growing nearby, [a young oak]]

I'd be inclined to saw it down beyond that split - I know that only leaves about a foot, but you can cut these trees to the ground and grow them as shrubs if you want - and now is the time to cut them.

I suppose I may have to do that but will be sorry to lose it and the privacy it provides. What is likely to happen if I just leave it ?

If it is fungus the disease will girdle the tree to a point where it will no longer be able to get nutrients and it will die.

Whereas if you cut it down, it will grow again, assuming there's no problem at the root. Eucalypts have strong apical domination, so a lead shoot will always develop and turn into a trunk eventually, if left. Eucalypts are fast growers - one cut to the ground in late March will be 6-7 feet high (as a bush) by September/October.

Also it matters where you want the privacy - left to its own devices, and assuming you have Eucalyptus gunii, these develop into a tree over 50 feet, and will confer no privacy below 10/15 feet which will all be the trunk, the leaf growth will all be higher up. Fine if you're screening yourself from a tall block of flats or the like, but not so useful if you just want privacy from a house at the back or next door.

Hi, if you cut these back annually, they will retain the juvenile foliage, which I personally prefer to the adult foliage, Derek.

Yea, me too,Derek - I always grow these as either 'lollipop' short trees, or shrubs. Keeping the lollipop shape isn't easy though...

Hi Bamboo, yes I agree, they grow far too fast for my liking, you can't take your eye off the ball with these, Derek.

It is supposed to be a Dalrympleana [according to the label when I bought it] but it doesn't have flaky bark like in the photos.
However I have seen a couple of photos of mature ones that appear to also have a vertical split up the trunk.
I would like to coppice it and will try to get someone to help me do it - on my own I can foresee a disaster !

Hi, if it is E dalrympleana, then this too will eventually get to about 70ft high with a spread of about 25 ft, but all the foregoing advice is still applicable, Derek.