Hi Autumn, two things: environmental protection, legal obligation.
To make sure you protect the environment, if the soil conditions are ok then you may be fine routing the grey water to the willow and alder patch. Even an information open wetland area may be sufficient to contain ALL of the effluent until it filters down through the tree roots. However, if you get runoff from this area, chances are it won't be nice for the local stream. If you get infiltration into the ground and there isn't enough depth of dry suitable soil, chances are it won't be nice for the groundwater.
Hence a filter of some sort. That needn't be sand though. It can be a constructed wetland, a gravel reed bed, a comfrey bed designed for both infiltration and nutrient capture… there's lots of options (www.wetlandsystems.ie/watertips shows more examples).
Legal obligation is an important consideration. Typically the aim of this is for environmental protection, so don't be too afraid to find out what codes you need to meet. This may be of some help if you haven't seen it already. If you don't meet the relevant codes you may be asked to undo hard work and may even be prosecuted, so do take care to find out what's needed, and to implement it. You needn't follow the most expensive option - just know what is required, that's all.
http://www.sepa.org.uk/about_us/publications/guidance/ppgs.aspx
https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/290141/pmho0706bjgl-e-e.pdf
Does that answer the question you have at all?