The following trees have a weeping style:

Atlas cedar
Cedrus atlantica
Loose diagonal branches with weeping twigs Evergreen
Cheal’s weeping cherry blossom tree
Prunus Kiku-shidare-zakura 
Twigs overhang in large weeping arches Deciduous
Deodar cedar
Cedrus deodara Pendula
Broad weeping with branches hanging to the ground Evergreen
Japanese pagoda tree
Sophora japonica Pendula
Broadly weeping crown Deciduous
Japanese weeping larch
Larix kaempferi Pendula
Broad weeping crown Deciduous
Golden weeping willow
Salix Chrysocoma
Broadly weeping golden branches hanging curtain-like to ground level Deciduous
Kilmarnock willow
Salix caprea Kilmarnock
Umbrella-shape crown Deciduous
Weeping Alaskan cedar
Chamaecyparis nootkatensis Pendula
Loose conical crown with arching branches Evergreen
Siberian violet-willow
Salix acutifolia Pendulifolia
Oval open crown, branches growing outwards and hanging down Deciduous
Weeping ash
Fraxinus excelsior Pendula
Wide arching weeping branches Deciduous
Weeping beech
Fagus sylvatica Pendula
Wide weeping crown Deciduous
Weeping elm
Ulmus glabra Pendula
Broad fan-shaped crown Deciduous
Weeping silver birch
Betula pendula Tristis 
Narrow weeping crown Deciduous
Weeping willow
Salix babylonica
Broad weeping branches drooping down to ground level Deciduous
Youngs weeping birch
Betula pendula Youngii
Wide weeping crown Deciduous