removing a Lelandii tree

You don't say how big your tree is but it obviously is quite tall if it upsets your neighbour. The cost will depend on where you live, rough costing in Surrey to grub out the whole tree would be £5.00 per foot i.e a 20ft. tree would cost £100.00. Also you would be charged for tipping the rubbish. We pay £90.00 per ton in this area. I can't say what your charges would be, but that would be a fair price.

if the tree is dumped on commen land you can dump it for nothing as long as its all bio dergradeable.i would just cut it down a little at a time from the top first .

We cut ours out ourselve's, as john said, from the top, a few feet at a time, we cut them down to ground level, then the hub took them to the council tip, theres a garden waste skip there..

a lump of dinamite maybe

Yola,
be very careful not to get exploited re costs. An elderly member of our family who can no longer garden much herself was told ridiculous sums for removing a large tree, but luckily she had the sense to get several quotes to compare, and get recommendations from friends or neighbours who'd had similar work done. I know this is a hassle, but worth it in the end.
If you really can't do the job gradually on your own, or with friends/family helping, you can at least save on the final bill by not having the felled wood removed.
What we did was advertise on our local freecycle group, and lots of people were happy to come and pick up the wood to burn (with conifer allow at least 9 months drying time before using it on a fire indoors) - more than one person came with their own chainsaw as I only had a handsaw and the last wood of the trunk was too much for me to cut. The green bits and smaller wood can be taken to tip or gradually cut up and put in your green council bin.

Sorry for overlong reply, hope it's some use to you. If you lived near me, I'd come and help - not sure where you are in the UK though!
I enjoy tree demolition jobs, slow and steady with handsaw - see photos on my homepage! And look up:
www.freecycle.org

We had many leylandii trees removed earlier this year.
We paid someone to chop the trees down and remove (they had an industrial shredder so imagine they used the chippings elsewhere). To keep the costs down we did not ask for the stumps / roots to be dug up or grinded - and as I have planted around the stumps with shrubs, they will soon be hidden anyway. I have also found previously that after a year or so, the stumps / roots tend to rot away naturally anyway, and until then I have useful 'steps' to balance on for weeding etc.
Chopping them down was the best decision we made garden wise, the hours of extra sunlight - bliss!