What plant will grow under a willow tree

Is this a weeping willow or another type? 

I think I'd go for spring bulbs, the wild-type daffodil, narcissus pseudo-narcissus would love it there, and if the beck floods a little in the winter that won't do the daffs any harm at all.

Then Lady's Mantle (alchemilla mollis) will grow in the shade of the tree and again won't worry if it gets wet feet occasionally and it's lovely pleated leaves decorated with raindrops, and its frothy acid green/yellow flowers will complement the light green of the willow leaves and the reddish colour of the twigs.  You can cut the alchemilla back in the late autumn so that in the spring when the daffs grow they have the scene to themselves 

cyclamen as well, the spring varieties would look stunnning.  Difficult place as the tree is taking so much nutrition from the area.  You could use pots or troughs there, place them when they are at their best and take them away for a rest when they are over, replacing with others. Alot of work though, for which you may not have time. 

You'd need to keep the hanging branches trimmed to at least head height in order to let in sufficient light for anything that's of interest after the spring bulbs.  You could try adding snake's head fritillaries to the bulbs as they like damp meadow conditions.

You could try variegated forms of ajuga reptans and pachysandra to give ground cover with some creamy highlights and also variegated vinca major.  This and the ajuga will give blue flowers and the pachysandra has white flowers.  Another plant worth a try would be pulmonaria which has spotted or silvered leaves depending on variety and white, pink or blue flowers - again depending on variety.   All of these will flower in spring or early summer but will give evergreen foliage interest, especially if you cut back the pulmonaria once flowering finishes so it produces new leaves.

A friend of mine puts tubs of busy lizzies under her tree as they appreciate the shade in summer.   This could be a plan to extend floral interest through the summer.

believe me I planted a willow from a twig it grew into a magnificent tree and shaded everything out .It had to come down .If you have a small weeping willow cyclamens are nice.