That's a good idea of @B3's . Then even if cuttings from your tree don't take, you can mix that bit of soil into your new garden and plant a new tree in memory of Snowy. Maybe an Amelianchier, one of its common names is snowy mespilus.
Try not to worry about what the new owners might or might not do. It's out of your control so no point in even thinking about it.
Doncaster, South Yorkshire. Soil type: sandy, well-drained
If you haven't had much rain recently it would be a good idea to give the tree a really good water the night before so that the shoots are as full of moisture as they can be, and take the cuttings in the early morning if possible.
Doncaster, South Yorkshire. Soil type: sandy, well-drained
Willows root very easily just in water so take cuttings and just leave them in a glass of water, changing the water ever few days, and I'm sure by the time you move there will be plenty of cuttings to come with you.
A potential problem with cuttings is that the tree is grafted onto a dwarfing rootstock to limit the growth. A cutting is likely to develop to the size of a 'normal' willow. No harm in trying though.
I still would try to move the tree. As you said, you don’t know if the new owner would cut it down anyway. Trees are robust and survive all kind of difficult conditions - this one is young and healthy, why shouldn’t it cope.
Have you checked the ground around the tree, it might be so compacted that it would be just too hard to even dig it out with big enough root ball, in which case you won’t have a choice but to leave it behind and that might make it easier to accept that you will move without it.
Take cuttings then have a go at moving it. There are a few ways you can do it, the most effective way is to dig a trench around the tree roughly in line with the canopy, then fill it with sand to get some root growth going and wait a bit, 6 weeks might be ok. If you can't do this get a good spade and dig it out, take your time, you can go through some roots but not lots, it will take some time to get it dug around so perhaps add some lose compost onto the roots to keep it drying out, keep this wet. Try and keep as much dirt on the rootball as you can. When it's out wrap it in sacking with more dirt around it. Keep it wet. It will weigh a ton so you'll need help. Ideally you will already have a space picked out at your new place and a suitable size hole dug. When you get it in the new hole back fill it with the dirt you took out of the hole, stake it as best you can to reduce movement, more than you would a new tree really. When it's in keep it very well watered and keep it watered, buckets not from a hose. If it takes its a win, if not you still have the cuttings. I've moved a Rowan 6 years in the ground in May and it's fine now. I've also got a twisted willow that my brother pulled out of his garden and left on the side for a day that is doing well. Water is the key.
The other thing to remember is that the fact that you care so much about Snowy's memory tells me that in life you must of loved and cared for them even more when they were with you, they must have loved their life because of you and there's nothing more you can do for a dog than love it, so don't be sad when you think of Snowy because they were never sad with you.
@TenN, would you care to estimate the weight of the large, saturated rootball + tree? I would think at least a cwt, maybe more. How do you lift such a weight and what sized plant pot do you advise buying in advance of the move?
Ben, I'm not sure if you're one of those men who like to ask rhetorical questions in the hope it lends them a sense of intellectual heft or you genuinely don't know the answer but I'll go with the latter because the first would make you an insufferable bore and I refuse to believe that about you. I wouldn't use a container if I had it wrapped and was keeping it wet, also why I suggested having a hole dug before it's taken out. I didn't for a second suggest that this can be done as a solitary endeavour, assumed that the OP would get some help. Hope that clears it up for you.
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The other thing to remember is that the fact that you care so much about Snowy's memory tells me that in life you must of loved and cared for them even more when they were with you, they must have loved their life because of you and there's nothing more you can do for a dog than love it, so don't be sad when you think of Snowy because they were never sad with you.