The following trees have a weeping style:

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